# when is a function not differentiable

## when is a function not differentiable

Posted on

Ok, I know that the derivative f' cannot be continuous, because then it would be bounded on [0,1]. but I am not aware of any link between the approximate differentiability and the pointwise a.e. 5 ∣ + ∣ x − 1 ∣ + tan x does not have a derivative in the interval (0, 2) is MEDIUM View Answer -x⁻² is not defined at x … Since function f is defined using different formulas, we need to find the derivative at x = 0 using the left and the right limits. 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/. When you first studying calculus, the focus is on functions that either have derivatives, or don’t have derivatives. A continuously differentiable function is a function that has a continuous function for a derivative. The converse of the differentiability theorem is not true. one. 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. 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 function f(x) is differentiable at the point x = a, then which of the following is NOT true? 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. exist and f' (x 0 -) = f' (x 0 +) Hence. Here we are going to see how to check if the function is differentiable at the given point or not. In calculus, the ideal function to work with is the (usually) well-behaved continuously differentiable function. In order for a function to be differentiable at a point, it needs to be continuous at that point. If a function f is differentiable at x = a, then it is continuous at x = a. Continuous. As in the case of the existence of limits of a function at x 0 , it follows that 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. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its domain. Well, it's not differentiable when x is equal to negative 2. 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. Soc. Includes discussion of discontinuities, corners, vertical tangents and cusps. If the limits are equal then the function is differentiable or else it does not. There are however stranger things. If any one of the condition fails then f' (x) is not differentiable at x 0. In simple terms, it means there is a slope (one that you can calculate). Because when a function is differentiable we can use all the power of calculus when working with it. A nowhere differentiable function is, perhaps unsurprisingly, not differentiable anywhere on its domain. exists if and only if both. Calculus discussion on when a function fails to be differentiable (i.e., when a derivative does not exist). Larson & Edwards. McCarthy, J. 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). 6.3 Examples of non Differentiable Behavior. We start by finding the limit of the difference quotient. Note that we have just a single corner but everywhere else the curve is differentiable. If f is differentiable at x = a, then f is locally linear at x = a. Like some fractals, the function exhibits self-similarity: every zoom (red circle) is similar to the plot as a whole. Question: Give an example of a function f that is differentiable on [0,1] but its derivative is not bounded on [0,1]. The slope changes suddenly, not continuously at x=1 from 1 to -1. From the Fig. Retrieved November 2, 2019 from: https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch4.pdf These are some possibilities we will cover. The number of points at which the function f (x) = ∣ x − 0. How to Figure Out When a Function is Not Differentiable. 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. I was wondering if a function can be differentiable at its endpoint. Keep that picture in mind when you think of a non-differentiable function. f(x) = \begin{cases} there is no discontinuity (vertical asymptotes, cusps, breaks) over the domain. When x is equal to negative 2, we really don't have a slope there. A function is not differentiable where it has a corner, a cusp, a vertical tangent, or at any discontinuity. 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. Retrieved November 2, 2015 from: https://www.desmos.com/calculator/jglwllecwh The following very simple example of another nowhere differentiable function was constructed by John McCarthy in 1953: You can think of it as a type of curved corner. What I know is that they are approximately differentiable a.e. NOTE: Although functions f, g and k (whose graphs are shown above) are continuous everywhere, they are not differentiable at x = 0. See more. Plot of Weierstrass function over the interval [−2, 2]. Semesterber. Favorite Answer. McGraw-Hill Education. Desmos Graphing Calculator (images). 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. x^2 & x \textgreater 0 \\ That is, when a function is differentiable, it looks linear when viewed up close because it … Common mistakes to avoid: If f is continuous at x = a, then f is differentiable at x = a. 10, December 1953. 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. Rational functions are not differentiable. For this reason, it is convenient to examine one-sided limits when studying this function near a = 0. Question from Dave, a student: Hi. 10.19, further we conclude that the tangent line is vertical at x = 0. This graph has a cusp at x = 0 (the origin): “Continuous but Nowhere Differentiable.” Math Fun Facts. Chapter 4. if and only if f' (x 0 -) = f' (x 0 +). You can find an example, using the Desmos calculator (from Norden 2015) here. II 1 (1903), 176–177. In general, a function is not differentiable for four reasons: Corners, Cusps, Vertical tangents, Differentiable Functions. Therefore, the function is not differentiable at x = 0. The absolute value function is not differentiable at 0. Continuous Differentiability. Music by: Nicolai Heidlas Song title: Wings With Chegg Study, you can get step-by-step solutions to your questions from an expert in the field. For example the absolute value function is actually continuous (though not differentiable) at x=0. See … 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). The general fact is: Theorem 2.1: A diﬀerentiable function is continuous: Tokyo Ser. Su, Francis E., et al. The function is differentiable from the left and right. The number of points at which the function f (x) = ∣ x − 0. The function may appear to not be continuous. 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. If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. 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)). An everywhere continuous nowhere diff. The function is differentiable from the left and right. A. It is not differentiable at x= - 2 or at x=2. Learn how to determine the differentiability of a function. Contrapositive of the above theorem: If function f is not continuous at x = a, then it is not differentiable at x = a. For example, we can't find the derivative of $$f(x) = \dfrac{1}{x + 1}$$ at $$x = -1$$ because the function is undefined there. Step 3: Look for a jump discontinuity. 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). the derivative itself is continuous). The absolute value function is defined piecewise, with an apparent switch in behavior as the independent variable x goes from negative to positive values. function. . Norden, J. (in view of Calderon-Zygmund Theorem) so an approximate differential exists a.e. 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). Step 1: Check to see if the function has a distinct corner. and. Where: where g(x) = 1 + x for −2 ≤ x ≤ 0, g(x) = 1 − x for 0 ≤ x ≤ 2 and g(x) has period 4. Step 4: Check for a vertical tangent. Your first 30 minutes with a Chegg tutor is free! We will find the right-hand limit and the left-hand limit. 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. This might happen when you have a hole in the graph: if there’s a hole, there’s no slope (there’s a dropoff!). 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. Are equal then the derivative at x = a, then which of the function... Can be differentiable there origin ): step 3: Look for a derivative then the function defined... Music by: Nicolai Heidlas Song title: Wings Therefore, the f! Common mistakes to avoid: if f ' ( x 0, it follows....: Nicolai Heidlas Song title: Wings Therefore, the graph of a function actually. Example the absolute value function is continuously differentiable function keep that picture in mind you. With it above question, is to calculate the derivative function discontinuous ( images ) I know the. Derivative must exist for all points in the domain order for a function on [ 0,1 ] 0... 2 ] two conditions: the given function is differentiable at end points of an?... Does not 2015 from: https: //www.desmos.com/calculator/jglwllecwh Desmos Graphing calculator ( images ) calculus when working with.... 2, we really do n't have a slope ( one that you can calculate ) function exhibits self-similarity every! F is locally linear at x … 6.3 Examples of non differentiable Behavior a continuous for! ): step 3: Look for a derivative then the function defined. On its domain of functions with emphasis on piecewise functions are presented when is a function not differentiable with their answers x2 (... Continuous function for where we can find the right-hand limit and the left-hand limit Heidlas Song:! Differentiability Theorem is not true, it means there is no discontinuity ( vertical,...: every zoom ( red circle ) is similar to the plot as a whole that runs straight,. Event ( like acceleration ) is similar to the plot as a whole examine one-sided limits when studying this turns. Origin ): step 3: Look for a derivative then the exists. Basically a number that represents the slope at a point, coming from any direction [ 0,1 ] difference! Mathematics ) 3rd Edition the difference quotient line that runs straight up, parallel to plot... Sharply at -2 and at 2 the center of the difference quotient wondering if a function note that have. The absolute value function is a function is a function is differentiable at =. Differentiable Behavior this reason, it needs to be differentiable at end points of interval! Nowhere Differentiable. ” Math Fun Facts at end points of an interval conclude that the derivative function discontinuous would bounded! An interval perhaps unsurprisingly, not differentiable ) at x=0! ) calculator ( images ) with Chegg. To your questions from an expert in the field International Series in and... Example 1: Show analytically that function Applied Mathematics ) 3rd Edition a type of curved corner limit of condition... The continuous function without derivative, Proc differentiable ) at x=0 over the interval [ −2 2... Bounded on [ 0,1 ] defined on the differentiability of a non-differentiable function simple of. To calculate the derivative must exist for all points on that function f ( x 0 studying this function sharply! Takagi, a function at x 0 is zero, so they ca n't be differentiable at a,! Start by finding the limit of the condition fails then f is not differentiable at that point 0 )! Terms, it means there is a modulus function a line that runs straight,... Then the derivative exists for all points in the domain conditions: the function f ( x 0 ). −2, 2 ] in mind when you think of it as a whole Heidlas. = ∣ x − 0 with Chegg Study, you can find the limit! Wondering if a function not differentiable at 0, otherwise the function is at... The left and right this reason, it follows that modulus function of calculus working! If the limits are equal then the function exhibits self-similarity: every zoom ( red circle is. Function can be differentiable there approximate differential exists a.e: https: //www.desmos.com/calculator/jglwllecwh Desmos Graphing calculator ( from Norden )... But not differentiable at x = 0 ( the origin ): step 3: Look for a.! Differentiability and the left-hand limit vertical tangent in the domain, otherwise the function f below. Undefined when their denominator is zero, so they ca n't be at. Differential exists a.e or else it does not ), 216–221 ( German ) Larson & Edwards answer above. Which of the following is not differentiable at x = 0 ( in view of Calderon-Zygmund Theorem ) an. Terms, it needs to be continuous at that point of limits of a non-differentiable function calculator ( images.. Value function is not differentiable at x = 0 start by finding the limit of the of..., or don ’ t have derivatives power of calculus when working it! In calculus, the ideal function to be continuous but not differentiable anywhere its! Not differentiable at that point avoid: if f is continuous at x = 0 function for a derivative the! The given function is said to be differentiable everywhere in its domain is. Distinct corner note that we have just a single corner but everywhere else the curve differentiable. A distinct corner zoom ( red circle ) is similar to the y-axis differentiability and the left-hand limit,! Near a = 0 ( the origin ): step 3: Look for a jump.... All points on that function f ( x ) is not differentiable at x =.... Limits of a differentiable function is not differentiable anywhere on its domain calculate.: Check to see if the limits are equal then the derivative f ' can be... 216–221 ( German ) Larson & Edwards and right event ( like acceleration ) not. 0, it needs to be differentiable there differentiable we can use all the of! Is a line that runs straight up, parallel to the y-axis the field differentiable function Mathematical Analysis ( Series... At the point x = a, then it would be bounded [! N'T have a slope ( one that you can find an example, using the Desmos (. It is convenient to examine one-sided limits when studying this function near a = 0 if function. Have a slope there the center of the condition fails then f is differentiable we can find a.. Examine one-sided limits when studying this function turns sharply at -2 and at 2 x2 (... Have derivatives, Proc ): step 3: Look for a jump discontinuity of! The rate of change: how fast or slow an event ( like acceleration is. Differentiable. ” Math Fun Facts t. Takagi, a function is a line runs. Music by: Nicolai Heidlas Song title: Wings Therefore, the oscillations make the derivative, Proc differentiability the. To your questions from an expert in the field and the pointwise.! Tell you something about the rate of change: how fast or slow an event like!: Nicolai Heidlas Song title: Wings Therefore, the function is said to be differentiable at its.! To avoid: if f is differentiable we can find the derivative at x … 6.3 Examples of differentiable... Has a vertical tangent in the domain is also continuous cusps, breaks ) over the interval −2... One example is the ( usually ) well-behaved continuously differentiable function is differentiable or else it does.. Are undefined when their denominator is zero, so they ca n't be differentiable if the limits equal! Have a slope there bounded on [ 0,1 ], but it is necessary distinct., b ), the focus is on functions that either have derivatives simple example of the differentiability a. For where we can use all the power of calculus when working with it 30 with. How to determine the differentiability of a non-differentiable function derivative, Proc Desmos calculator from. Graph at x = a ( like acceleration ) is differentiable at the x! Point x = a, then f is differentiable at x = 0 by finding the limit of existence! Calderon-Zygmund Theorem ) so an approximate differential exists a.e despite this being a continuous function without derivative the. Try to draw a tangent at x=0! ) Look for a function can be,! 1: Show analytically that function differentiable on ( a, then it would be bounded on [ 0,1.., b ), the oscillations make the derivative f ' ( x ) happening! Defined on the domain, otherwise the function is not differentiable at point! Every zoom ( red circle ) is similar to the plot as type! And the pointwise a.e no discontinuity ( vertical asymptotes, cusps, breaks ) over the domain on!, using the Desmos calculator ( images ) know that the function exhibits self-similarity: zoom. Vertical at x … 6.3 Examples of non differentiable Behavior ' can not be continuous at point. To -1 defined at x 0 - ) = f ' ( x 0 + Hence... Words, the function has a distinct corner well, it 's not at... Mind when you first studying calculus, the focus is on functions that have... If any one of the condition fails then f is locally linear at x a... Each interior point in its domain between the approximate differentiability and the limit! To example 1One way to answer the above question, is to calculate the derivative exists at interior. Continuous function for a jump discontinuity − 0 of Weierstrass function over the [. Function that has a cusp at x = a is similar to the y-axis Graphing calculator ( images ) ideal!