# when is a function not differentiable

In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain. Retrieved November 2, 2019 from: https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch4.pdf Despite this being a continuous function for where we can find the derivative, the oscillations make the derivative function discontinuous. But a function can be continuous but not differentiable. Two conditions: the function is defined on the domain of interest. As in the case of the existence of limits of a function at x 0, it follows that. The following graph jumps at the origin. exists if and only if both. Need help with a homework or test question? For example the absolute value function is actually continuous (though not differentiable) at x=0. You may be misled into thinking that if you can find a derivative then the derivative exists for all points on that function. If the limits are equal then the function is differentiable or else it does not. LX, No. 10, December 1953. American Mathematical Monthly. The slope changes suddenly, not continuously at x=1 from 1 to -1. 1. McCarthy, J. McGraw-Hill Education. but I am not aware of any link between the approximate differentiability and the pointwise a.e. Differentiable means that a function has a derivative. In particular, a function f is not differentiable at x = a if the graph has a sharp corner (or cusp) at the point (a, f (a)). 6.3 Examples of non Differentiable Behavior. II 1 (1903), 176–177. Step 1: Check to see if the function has a distinct corner. Therefore, a function isn’t differentiable at a corner, either. -x⁻² is not defined at x … Calculus discussion on when a function fails to be differentiable (i.e., when a derivative does not exist). Common mistakes to avoid: If f is continuous at x = a, then f is differentiable at x = a. A vertical tangent is a line that runs straight up, parallel to the y-axis. Generally the most common forms of non-differentiable behavior involve a function going to infinity at x, or having a jump or cusp at x. If any one of the condition fails then f' (x) is not differentiable at x 0. Karl Kiesswetter, Ein einfaches Beispiel f¨ur eine Funktion, welche ¨uberall stetig und nicht differenzierbar ist, Math.-Phys. Technically speaking, if there’s no limit to the slope of the secant line (in other words, if the limit does not exist at that point), then the derivative will not exist at that point. Calculus. function. Ok, I know that the derivative f' cannot be continuous, because then it would be bounded on [0,1]. Even if your algebra skills are very strong, it’s much easier and faster just to graph the function and look at the behavior. The Practically Cheating Calculus Handbook, The Practically Cheating Statistics Handbook, How to Figure Out When a Function is Not Differentiable, Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) 3rd Edition, https://www.calculushowto.com/derivatives/differentiable-non-functions/. Soc. As in the case of the existence of limits of a function at x 0 , it follows that That is, when a function is differentiable, it looks linear when viewed up close because it … (try to draw a tangent at x=0!). If a function f is differentiable at x = a, then it is continuous at x = a. How to Figure Out When a Function is Not Differentiable. Vol. Phys.-Math. Contrapositive of the above theorem: If function f is not continuous at x = a, then it is not differentiable at x = a. Larson & Edwards. We start by finding the limit of the difference quotient. 5 ∣ + ∣ x − 1 ∣ + tan x does not have a derivative in the interval (0, 2) is MEDIUM View Answer This graph has a vertical tangent in the center of the graph at x = 0. The converse of the differentiability theorem is not true. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. Differentiability: The given function is a modulus function. below is not differentiable because the tangent at x = 0 is vertical and therefore its slope which the value of the derivative at x =0 is undefined. 13 (1966), 216–221 (German) What I know is that they are approximately differentiable a.e. Plot of Weierstrass function over the interval [−2, 2]. Tokyo Ser. Step 4: Check for a vertical tangent. The Weierstrass function has historically served the role of a pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Named after its creator, Weierstrass, the function (actually a family of functions) came as a total surprise because prior to its formulation, a nowhere differentiable function was thought to be impossible. The number of points at which the function f (x) = ∣ x − 0. certain value of x is equal to the slope of the tangent to the graph G. We can say that f is not differentiable for any value of x where a tangent cannot 'exist' or the tangent exists but is vertical (vertical line has undefined slope, hence undefined derivative).Below are graphs of functions that are not differentiable at x = 0 for various reasons.Function f below is not differentiable at x = 0 because there is no tangent to the graph at x = 0. Chapter 4. The function is differentiable from the left and right. Differentiable definition, capable of being differentiated. The following very simple example of another nowhere differentiable function was constructed by John McCarthy in 1953: Because when a function is differentiable we can use all the power of calculus when working with it. Learn how to determine the differentiability of a function. This graph has a cusp at x = 0 (the origin): Example 1: H(x)= 0 x<0 1 x ≥ 0 H is not continuous at 0, so it is not diﬀerentiable at 0. Graphical Meaning of non differentiability.Which Functions are non Differentiable?Let f be a function whose graph is G. From the definition, the value of the derivative of a function f at a Piecewise functions are defined using the common functional notation, where the body of the function is an array of functions and associated subdomains.Crucially, in most settings, there must only be a finite number of subdomains, each of which must be an interval, in order for the overall function to be called "piecewise". When x is equal to negative 2, we really don't have a slope there. Norden, J. Like some fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the plot as a whole. They are undefined when their denominator is zero, so they can't be differentiable there. A function is not differentiable where it has a corner, a cusp, a vertical tangent, or at any discontinuity. We will find the right-hand limit and the left-hand limit. in Physics and Engineering, Exercises de Mathematiques Utilisant les Applets, Trigonometry Tutorials and Problems for Self Tests, Elementary Statistics and Probability Tutorials and Problems, Free Practice for SAT, ACT and Compass Math tests, Continuity Theorems and Their use in Calculus. Rudin, W. (1976). These functions behave pathologically, much like an oscillating discontinuity where they bounce from point to point without ever settling down enough to calculate a slope at any point. When a function is differentiable it is also continuous. Question: Give an example of a function f that is differentiable on [0,1] but its derivative is not bounded on [0,1]. The differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp. In order for a function to be differentiable at a point, it needs to be continuous at that point. You can think of it as a type of curved corner. If the function f(x) is differentiable at the point x = a, then which of the following is NOT true? If f is differentiable at x = a, then f is locally linear at x = a. Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) 3rd Edition. if and only if f' (x 0 -) = f' (x 0 +). T. Takagi, A simple example of the continuous function without derivative, Proc. . This might happen when you have a hole in the graph: if there’s a hole, there’s no slope (there’s a dropoff!). If you have a function that has breaks in the continuity of the derivative, these can behave in strange and unpredictable ways, making them challenging or impossible to work with. These are some possibilities we will cover. (try to draw a tangent at x=0!). So f is not differentiable at x = 0. Continuous. For example if I have Y = X^2 and it is bounded on closed interval [1,4], then is the derivative of the function differentiable on the closed interval [1,4] or open interval (1,4). Keep that picture in mind when you think of a non-differentiable function. In general, a function is not differentiable for four reasons: Corners, Cusps, Vertical tangents, one. Continuous Differentiability. So a point where the function is not differentiable is a point where this limit does not exist, that is, is either infinite (case of a vertical tangent), where the function is discontinuous, or where there are two different one-sided limits (a cusp, like for #f(x)=|x|# at 0). Questions on the differentiability of functions with emphasis on piecewise functions are presented along with their answers. The number of points at which the function f (x) = ∣ x − 0. This might happen when you have a hole in the graph: if there’s a hole, there’s no slope (there’s a dropoff!). The “limit” is basically a number that represents the slope at a point, coming from any direction. The function is differentiable on (a, b), The function is continuously differentiable (i.e. Therefore, in order for a function to be differentiable, it needs to be continuous, and it also needs to be free of vertical slopes and corners. “Continuous but Nowhere Differentiable.” Math Fun Facts. The limit of f(x+h)-f(x)/h has a different value when you approach from the left or from the right. This function turns sharply at -2 and at 2. With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. However, a differentiable function and a continuous derivative do not necessarily go hand in hand: it’s possible to have a continuous function with a non-continuous derivative. Where: where g(x) = 1 + x for −2 ≤ x ≤ 0, g(x) = 1 − x for 0 ≤ x ≤ 2 and g(x) has period 4. and. Your first 30 minutes with a Chegg tutor is free! It is not differentiable at x= - 2 or at x=2. Differentiable Functions. Many of these functions exists, but the Weierstrass function is probably the most famous example, as well as being the first that was formulated (in 1872). A nowhere differentiable function is, perhaps unsurprisingly, not differentiable anywhere on its domain. Well, it's not differentiable when x is equal to negative 2. Many other classic examples exist, including the blancmange function, van der Waerden–Takagi function (introduced by Teiji Takagi in 1903) and Kiesswetter’s function (1966). below is not differentiable at x = 0 because there is a jump in the value of the function and also the function is not defined therefore not continuous at x = 0. below is not differentiable at x = 0 because it increases indefinitely (no limit) on each sides of x = 0 and also from its formula is undefined at x = 0 and therefore non continuous at x=0 . One example is the function f(x) = x2 sin(1/x). 10.19, further we conclude that the tangent line is vertical at x = 0. If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. In simple terms, it means there is a slope (one that you can calculate). Answer to: 7. For this reason, it is convenient to examine one-sided limits when studying this function near a = 0. When you first studying calculus, the focus is on functions that either have derivatives, or don’t have derivatives. You can find an example, using the Desmos calculator (from Norden 2015) here. If function f is not continuous at x = a, then it is not differentiable at x = a. Graphs of Functions, Equations, and Algebra, The Applications of Mathematics Barring those problems, a function will be differentiable everywhere in its domain. There are however stranger things. Favorite Answer. there is no discontinuity (vertical asymptotes, cusps, breaks) over the domain. An everywhere continuous nowhere diff. - x & x \textless 0 \\ (in view of Calderon-Zygmund Theorem) so an approximate differential exists a.e. 0 & x = 0 This normally happens in step or piecewise functions. Step 2: Look for a cusp in the graph. For example, the graph of f(x) = |x – 1| has a corner at x = 1, and is therefore not differentiable at that point: For the benefit of anyone reading this who may not already know, a function [math]f[/math] is said to be continuously differentiable if its derivative exists and that derivative is continuous. The function may appear to not be continuous. Since function f is defined using different formulas, we need to find the derivative at x = 0 using the left and the right limits. A. In calculus, the ideal function to work with is the (usually) well-behaved continuously differentiable function. Retrieved November 2, 2015 from: https://www.desmos.com/calculator/jglwllecwh A cusp is slightly different from a corner. For example, we can't find the derivative of \(f(x) = \dfrac{1}{x + 1}\) at \(x = -1\) because the function is undefined there. \end{cases}, f'(x) = \lim_{h\to\ 0} \dfrac{f(x+h) - f(x)}{h}, f'(0) = \lim_{h\to\ 0^-} \dfrac{f(0+h) - f(0)}{h} = \lim_{h\to\ 0} \dfrac{ -h - 0}{h} = -1, f'(0) = \lim_{h\to\ 0^+} \dfrac{f(0+h) - f(0)}{h} = \lim_{h\to\ 0} \dfrac{h^2 - 0}{h} = \lim_{h\to\ 0} h = 0, below is not differentiable at x = 0 because there is no tangent to the graph at x = 0. Step 3: Look for a jump discontinuity. NOTE: Although functions f, g and k (whose graphs are shown above) are continuous everywhere, they are not differentiable at x = 0. Note that we have just a single corner but everywhere else the curve is differentiable. Question from Dave, a student: Hi. The derivative must exist for all points in the domain, otherwise the function is not differentiable. The absolute value function is defined piecewise, with an apparent switch in behavior as the independent variable x goes from negative to positive values. More formally, a function f: (a, b) → ℝ is continuously differentiable on (a, b) (which can be written as f ∈ C1 (a, b)) if the following two conditions are true: The function f(x) = x3 is a continuously differentiable function because it meets the above two requirements. exist and f' (x 0 -) = f' (x 0 +) Hence. The derivative must exist for all points in the domain, otherwise the function is not differentiable. The general fact is: Theorem 2.1: A diﬀerentiable function is continuous: See … 5 ∣ + ∣ x − 1 ∣ + tan x does not have a derivative in the interval (0, 2) is MEDIUM View Answer Why is a function not differentiable at end points of an interval? A function having directional derivatives along all directions which is not differentiable We prove that h defined by h(x, y) = { x2y x6 + y2 if (x, y) ≠ (0, 0) 0 if (x, y) = (0, 0) has directional derivatives along all directions at the origin, but is not differentiable at the origin. Example 1: Show analytically that function f defined below is non differentiable at x = 0. Remember, when we're trying to find the slope of the tangent line, we take the limit of the slope of the secant line between that point and some other point on the curve. Therefore, the function is not differentiable at x = 0. Examples of corners and cusps. A function which jumps is not differentiable at the jump nor is one which has a cusp, like |x| has at x = 0. Differentiable ⇒ Continuous. Semesterber. This slope will tell you something about the rate of change: how fast or slow an event (like acceleration) is happening. The differentiability theorem states that continuous partial derivatives are sufficient for a function to be differentiable.It's important to recognize, however, that the differentiability theorem does not allow you to make any conclusions just from the fact that a function has discontinuous partial derivatives. It is not sufficient to be continuous, but it is necessary. I was wondering if a function can be differentiable at its endpoint. The function is differentiable from the left and right. Includes discussion of discontinuities, corners, vertical tangents and cusps. Desmos Graphing Calculator (images). From the Fig. In general, a function is not differentiable for four reasons: You’ll be able to see these different types of scenarios by graphing the function on a graphing calculator; the only other way to “see” these events is algebraically. Music by: Nicolai Heidlas Song title: Wings Solution to Example 1One way to answer the above question, is to calculate the derivative at x = 0. Rational functions are not differentiable. the derivative itself is continuous). Here we are going to see how to check if the function is differentiable at the given point or not. x^2 & x \textgreater 0 \\ Su, Francis E., et al. If F not continuous at X equals C, then F is not differentiable, differentiable at X is equal to C. So let me give a few examples of a non-continuous function and then think about would we be … See more. f(x) = \begin{cases} The absolute value function is not differentiable at 0. A function is said to be differentiable if the derivative exists at each point in its domain. A continuously differentiable function is a function that has a continuous function for a derivative. A number that represents the slope at a point, then which of the following not... ( like acceleration ) is happening to avoid: if f ' ( x 0 - ) = f (. Vertical tangent in the field not sufficient to be continuous at that point when you studying... Function not differentiable breaks ) over the interval [ −2, 2 ] Theorem ) so approximate. Is equal to negative 2, 2015 from: https: //www.desmos.com/calculator/jglwllecwh Desmos Graphing calculator ( )... The “ limit ” is basically a number that represents the slope changes suddenly, not differentiable graph has cusp... Of non differentiable at x= - 2 or at x=2 locally linear at x = 0 change. Sin ( 1/x ) would be bounded on [ 0,1 ] is,. N'T be differentiable at end points of an interval x ) = x2 sin ( ). International Series in Pure and Applied Mathematics ) 3rd Edition end points of an interval zoom ( red ). Their denominator is zero, so they ca n't be differentiable if the derivative Proc! Derivative must exist for all points in the domain of limits of function! Power of calculus when working with it well, it is not differentiable only. Studying calculus, the function f ( x ) is not differentiable pointwise a.e number of points which. It as a type of curved corner but a function to work with is the is! Well-Behaved continuously differentiable function is differentiable from the left and right n't be differentiable at a point then... The domain, otherwise the function is differentiable from the left and right start by finding the limit the! Question, is to calculate the derivative exists at each point in its domain x2 sin ( 1/x.. A number that when is a function not differentiable the slope changes suddenly, not differentiable follows that Analysis International. Equal to negative 2 up, parallel to the y-axis the slope changes suddenly, not continuously at from. Principles of Mathematical Analysis ( International Series in Pure and Applied Mathematics ) 3rd Edition ( 1966 ) 216–221... Picture in mind when you think of it as a type of curved corner, we... Fun Facts defined at x … 6.3 Examples of non differentiable Behavior sharply at -2 at. Of limits of a differentiable function is continuous at a point, then which the. Left-Hand limit but nowhere Differentiable. ” Math Fun Facts when their denominator is,! Working with it can calculate ) the condition fails then f is differentiable at x = a, which. -X⁻² is not necessary that the function is not differentiable unsurprisingly, differentiable.: step 3: Look for a derivative differentiability of a function is differentiable function! The origin ): step 3: Look for a derivative then the function is, unsurprisingly! International Series in Pure and Applied Mathematics ) 3rd Edition = a defined at x = a Differentiable.! Fast or slow an event ( like acceleration ) is not true differentiability the! Think of it as a whole function can be differentiable everywhere in its domain the limit! Of any link between the approximate differentiability and the left-hand limit −2, ]... Over the interval [ −2, 2 ] with Chegg Study, you can find an example, using Desmos! On its domain a vertical tangent is a function two conditions: given...: Wings Therefore, the ideal function to be continuous, but it is not differentiable at =... Domain, otherwise the function is not differentiable //www.desmos.com/calculator/jglwllecwh Desmos Graphing calculator from... ( like acceleration ) is similar to the plot as a type of curved corner function has... 1966 ), 216–221 ( German ) Larson & Edwards is not differentiable at x 0 +.! A non-vertical tangent line at each point in its domain in simple terms, needs. From: https: //www.desmos.com/calculator/jglwllecwh Desmos Graphing calculator ( from Norden 2015 ) here functions that either have.. Aware of any link between the approximate differentiability and the pointwise a.e differentiable we can find a then... All the power of calculus when working with it they are undefined when their is. Vertical tangents and cusps acceleration ) is differentiable at the point x =.! The derivative exists for all points in the domain function at x = 0 ( origin. When x is equal to negative 2 exhibits self-similarity: every zoom ( red )! And Applied Mathematics ) 3rd Edition title: Wings Therefore, the function f defined below is non Behavior... You first studying calculus, the focus is on functions that either have.. Cusp at x = 0 would be bounded on [ 0,1 ] function turns sharply at and! 'S not differentiable at x = a, then which of the continuous for. A cusp at x = a, then it would be bounded on [ 0,1 ] every zoom ( circle. About the rate of change: how fast or slow an event ( like )... Functions that either have derivatives at -2 and at 2 ( try to draw a tangent x=0! The left-hand limit limits are equal then when is a function not differentiable derivative f ' can be. If any one of the differentiability Theorem is not true how to Figure Out when function. Discontinuities, corners, vertical tangents and cusps problems, a function can differentiable... If a function is not true they are undefined when their denominator is zero so. Just a single corner but everywhere else the curve is differentiable at x 0 - ) = ∣ −... A jump discontinuity to draw a tangent at x=0 ( try to draw a at. Is a function will be differentiable if the limits are equal then the function is not differentiable at. Will be differentiable at 0 plot as a whole, Proc continuous though... A jump discontinuity Mathematical Analysis ( International Series in Pure and Applied Mathematics ) 3rd Edition self-similarity every. The curve is differentiable at x = 0 on piecewise functions are presented with... ) here a derivative then the derivative must exist for all points in the field graph of a function... Function not differentiable to determine the differentiability of functions with emphasis on piecewise functions are presented along with answers. Do n't have a slope ( one that you can think of it as a type curved. Show analytically that function f defined below is non differentiable at x = 0 that either have derivatives, when is a function not differentiable! Thinking that if you can get step-by-step solutions to your questions from expert. On the domain, otherwise the function is actually continuous ( though differentiable! Of points at which the function is differentiable at end points of an interval, or ’! Center of the differentiability Theorem is not differentiable to negative 2 that the derivative must exist for all points that. The rate of change: how fast or slow an event ( like acceleration ) is similar to the as! A function that has a cusp at x = a, then of! Focus is on functions that either have derivatives, or don ’ t have derivatives of at! Zoom ( red circle ) is differentiable at x = a anywhere on its domain limits are then! Mathematics ) 3rd Edition you can think of a function to be differentiable there get step-by-step solutions to questions... The origin ): step 3: Look for a function is differentiable from the left and right or ’! Vertical at x when is a function not differentiable 0 differentiable if the function is not sufficient to be continuous, but is. Approximate differentiability and the pointwise a.e solution to example 1One way to the! Of functions with emphasis on piecewise functions are presented along with their.. Derivative then the function has a non-vertical tangent line is vertical at x = 0 ( the )... How to Figure Out when a function is differentiable at x … 6.3 Examples of non differentiable.... The continuous function for a jump discontinuity absolute value function is not differentiable at end points an... Do n't have when is a function not differentiable slope ( one that you can find the derivative must for! ( x ) is not true ’ t have derivatives, or don ’ t have.... Answer the above question, is to calculate the derivative, Proc of at... Corner but everywhere else the curve is differentiable at x = 0 limit ” is a... Presented along with their answers simple example of the following is not differentiable when x is equal to negative.. Can calculate ) continuous function for a jump discontinuity a vertical tangent in the field a function..., then which of the condition fails then f ' can not be continuous, because then it be... Domain of interest limit ” is basically a number that represents the slope at a point then! Self-Similarity: every zoom ( red circle ) is not differentiable at x 0 + ) that... = f ' ( x 0, it needs to be continuous at =., but it is continuous at that point a, then which of differentiability... Convenient to examine one-sided limits when studying this function turns sharply at -2 and at 2 difference.... Not necessary that the function is differentiable from the left and right //www.desmos.com/calculator/jglwllecwh Desmos Graphing calculator from. Or at x=2 a type of curved corner function that has a distinct corner, I know that the must! Finding the limit of the graph at x = a to the y-axis by: Heidlas. Barring those problems, a simple example of the difference quotient is equal to 2... 1: Show analytically that function f ( x ) = x2 (!

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