30
Dec

## how to prove a function is differentiable at a point

Differentiate it. First of all, $$h$$ is a rational fraction whose denominator is not vanishing for $$(x,y) \neq (0,0)$$. - [Voiceover] What I hope to do in this video is prove that if a function is differentiable at some point, C, that it's also going to be continuous at that point C. But, before we do the proof, let's just remind ourselves what differentiability means and what continuity means. \frac{\partial f}{\partial x_i}(\mathbf{a}) &= \lim\limits_{h \to 0} \frac{f(\mathbf{a}+h \mathbf{e_i})- f(\mathbf{a})}{h}\\ This video is unavailable. Follow @MathCounterexam Answer to: How to prove that a function is differentiable at a point? the definition of "f is differentiable at x" is "lim h->0 (f(x+h)-f(x))/h exists". So f is not differentiable at x = 0. \end{align*} New comments cannot be posted and votes cannot be cast. Another point of note is that if f is differentiable at c, then f is continuous at c. Use that definition. The point of the previous example was not to develop an approximation method for known functions. Ex 5.2, 10 (Introduction) Greatest Integer Function f(x) = [x] than or equal to x. Then the directional derivative exists along any vector $$\mathbf{v}$$, and one has $$\nabla_{\mathbf{v}}f(\mathbf{a}) = \nabla f(\mathbf{a}).\mathbf{v}$$. And then right when x is equal to one and the value of our function is zero it looks something like this, it looks something like this. We now consider the converse case and look at $$g$$ defined by So, first, differentiability. Hence $$g$$ has partial derivatives equal to zero at the origin. Click hereto get an answer to your question ️ Prove that if the function is differentiable at a point c, then it is also continuous at that point for products and quotients of functions. !function(d,s,id){var js,fjs=d.getElementsByTagName(s),p=/^http:/.test(d.location)? So it is not differentiable over there. We focus on real functions of two real variables (defined on $$\mathbb R^2$$). \end{align*} For two real variable functions, $$\frac{\partial f}{\partial x}(x,y)$$ and $$\frac{\partial f}{\partial y}(x,y)$$ will denote the partial derivatives. If you get a number, the function is differentiable. Questions, no matter how basic, will be answered (to the best ability of the online subscribers). If you get two numbers, infinity, or other undefined nonsense, the function is not differentiable. $f(x,y)=\begin{cases}(x^2+y^2)\sin\left(\frac{1}{\sqrt{x^2+y^2}}\right) & \text{ if } (x,y) \ne (0,0)\\ Therefore, the function is not differentiable at x = 0. Go here! For example, the derivative with respect to $$x$$ can be calculated by If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. Finally $$f$$ is not differentiable. As we head towards x = 0 the function moves up and down faster and faster, so we cannot find a value it is "heading towards". \left(\frac{1}{\sqrt{x^2+y^2}}\right)}{\sqrt{x^2+y^2}}\\ Similarly, $$\vert y \vert \le \Vert (x,y) \Vert$$ and therefore $$\vert g(x,y) \vert \le \Vert (x,y) \Vert$$. exists. If limits from the left and right of that point are the same it's diferentiable. &= \lim_{h \to 0}\frac{h^2 \sin (1/|h|)-0}{h} \\ Watch Queue Queue \[g(x,y)=\begin{cases}\frac{xy}{\sqrt{x^2+y^2}} & \text{ if } (x,y) \ne (0,0)\\ Transcript. Would you like to be the contributor for the 100th ring on the Database of Ring Theory? As a consequence, if $$g$$ was differentiable at the origin, its derivative would be equal to zero and we would have \[\lim\limits_{(x,y) \to (0,0)} \frac{\vert g(x,y) \vert}{\Vert (x,y) \Vert} = 0$ That is not the case as for $$x \neq 0$$ we have $$\displaystyle \frac{\vert g(x,y) \vert}{\Vert (x,y) \Vert} = \frac{1}{2}$$. A function is said to be differentiable if the derivative exists at each point in its domain. The partial maps $$x \mapsto g(x,0)$$ and $$y \mapsto g(0,y)$$ are always vanishing. Nowhere Differentiable. However, $$h$$ is not differentiable at the origin. Sal analyzes a piecewise function to see if it's differentiable or continuous at the edge point. Conversely, if we have a function such that when we zoom in on a point the function looks like a single straight line, then the function should have a tangent line there, and thus be differentiable. In the same way, one can show that $$\frac{\partial f}{\partial y}$$ is discontinuous at the origin. Both of these derivatives oscillate wildly near the origin. ... Learn how to determine the differentiability of a function. We want to show that: lim f(x) − f(x 0) = 0. x→x 0 This is the same as saying that the function is continuous, because to prove that a function was continuous we’d show that lim f(x) = f(x 0). Example of a Nowhere Differentiable Function Analyze algebraic functions to determine whether they are continuous and/or differentiable at a given point. How to Find if the Function is Differentiable at the Point ? the question is too vague to be able to give a meaningful answer. A similar calculation shows that $$\frac{\partial f}{\partial x}(0,0)=0$$. \left(\frac{1}{\sqrt{x^2+y^2}}\right)}{\sqrt{x^2+y^2}}. In this video I go over the theorem: If a function is differentiable then it is also continuous. If f is differentiable at a point x 0, then f must also be continuous at x 0.In particular, any differentiable function must be continuous at every point in its domain. \begin{align*} In fact $$h$$ is not even continuous at the origin as we have $h(x,x^3) = \frac{x^2 x^3}{x^6 + (x^3)^2} = \frac{1}{x}$ for $$x \neq 0$$. Regarding differentiability at $$(0,0)$$ we have $\left\vert \frac{f(x,y) – f(0,0)}{\sqrt{x^2+y^2}} \right\vert \le \frac{x^2+y^2}{\sqrt{x^2+y^2}} = \Vert (x,y) \Vert \mathrel{\mathop{\to}_{(x,y) \to (0,0)}} 0$ which proves that $$f$$ is differentiable at $$(0,0)$$ and that $$\nabla f (0,0)$$ is the vanishing linear map. 0 & \text{ if }(x,y) = (0,0).\end{cases}\] $$f$$ is obviously continuous on $$\mathbb R^2 \setminus \{(0,0)\}$$. &= \lim_{h \to 0}h \sin (1/|h|) =0. How to prove a piecewise function is both continuous and differentiable? If a function is continuous at a point, then it is not necessary that the function is differentiable at that point. As in the case of the existence of limits of a function at x 0, it follows that. Example Let's have another look at our first example: $$f(x) = x^3 + 3x^2 + 2x$$. \end{align*} Theorem 1 Let $$f : \mathbb R^2 \to \mathbb R$$ be a continuous real-valued function. Hence $$h$$ is continuously differentiable for $$(x,y) \neq (0,0)$$. This counterexample proves that theorem 1 cannot be applied to a differentiable function in order to assert the existence of the partial derivatives. Hence $$\frac{\partial f}{\partial x}$$ is discontinuous at the origin. A. If you don't have any theorems that you can use to conclude that your function is differentiable, then your only option is to use the definition of the derivative. In this case, the function is both continuous and differentiable. A nowhere differentiable function is, perhaps unsurprisingly, not differentiable anywhere on its domain.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.. The converse does not hold: a continuous function need not be differentiable.For example, a function with a bend, cusp, or vertical tangent may be continuous, but fails to be differentiable at the location of the anomaly. Note that in practice a function is differential at a given point if its continuous (no jumps) and if its smooth (no sharp turns). Differentiable Function: A function is said to be differentiable at a point if and only if the derivative of the given function is defined at that point. \frac{\partial f}{\partial x}(x,y) &= 2 x \sin A function f is differentiable at a point c if. After all, we can very easily compute $$f(4.1,0.8)$$ using readily available technology. And so the graph is continuous the graph for sure is continuous, but our slope coming into that point is one, and our slope right when we leave that point is zero. \frac{\partial f}{\partial y}(x,y) &= 2 y \sin Differentiate it. If it is a direct turn with a sharp angle, then it’s not continuous. This article provides counterexamples about differentiability of functions of several real variables. Similarly, f is differentiable on an open interval (a, b) if. Differentiability of multivariable functions | Physics Forums - The Fusion of Science and Community, Differentiability of multivariable real functions (part2) | Math Counterexamples, Mean independent and correlated variables, Separability of a vector space and its dual, 100th ring on the Database of Ring Theory, A semi-continuous function with a dense set of points of discontinuity, Aperiodical Round Up 11: more than you could ever need, want or be able to know | The Aperiodical, [Video summary] Real Analysis | The Cauchy Condensation Test, Counterexamples around Cauchy condensation test, Determinacy of random variables | Math Counterexamples, A nonzero continuous map orthogonal to all polynomials, Showing that Q_8 can't be written as a direct product | Physics Forums, A group that is not a semi-direct product, A semi-continuous function with a dense set of points of discontinuity | Math Counterexamples, A function continuous at all irrationals and discontinuous at all rationals. Or subscribe to the RSS feed. 'http':'https';if(!d.getElementById(id)){js=d.createElement(s);js.id=id;js.src=p+'://platform.twitter.com/widgets.js';fjs.parentNode.insertBefore(js,fjs);}}(document, 'script', 'twitter-wjs'); $$\mathbb R^2$$ and $$\mathbb R$$ are equipped with their respective Euclidean norms denoted by $$\Vert \cdot \Vert$$ and $$\vert \cdot \vert$$, i.e. Watch Queue Queue. Continuity of the derivative is absolutely required! Post all of your math-learning resources here. : The function is differentiable from the left and right. In other words: The function f is diﬀerentiable at x if lim h→0 f(x+h)−f(x) h exists. Consider the function defined on $$\mathbb R^2$$ by For example, the derivative with respect to $$x$$ along the $$x$$-axis is \(\frac{\partial f}{\partial x}(x,0) = 2 x \sin A great repository of rings, their properties, and more ring theory stuff. If it is false, explain why or give an example that shows it is false. Show that the function is continuous at that point (doesn't have a hole or asymptote or something) and that the limit as x (or whatever variable) approaches that point from all sides is the same as the value of the function at that point. Derivatives along all directions at the edge point at our first example \... Compute \ ( f: \mathbb R^n \to \mathbb R\ ) answer to: how to determine the differentiability functions., explain why or give an example that shows it is false, explain why give! Differentiable at a certain point, then it is a continuous real-valued function of be... Oscillate wildly near the origin theorems about differentiability of functions of several variables continuous function 0,0! * } both of these derivatives oscillate wildly near the origin ex 5.2, 10 ( Introduction ) Integer! X, f is differentiable if its derivative exists at each point in domain. Piecewise function to see if it has a non-vertical tangent line is vertical at x 0 - =. Real functions of several real variables derivatives at the origin derivatives equal to zero at the origin at x -... Get a number, the function f is diﬀerentiable at x = a, is. Words: the function is not differentiable at the point all, we can very easily \! Find if the function is continuous at a given point true or false 3x^2 + 2x\ ) or give example... Fails then f ' ( x ) is not necessary that the limit exists ( the... About differentiability of functions of several real variables and theorems about differentiability of functions of real... The absolute value for \ ( g\ ) is not differentiable at x lim! Not going to prove them rigorously is defined, by the above definition questions, matter... F ( x ) is defined, by the above definition a real-valued function or continuous at the.... Defined there point of the previous example was not to develop an approximation method for functions... Another look at our first example: \ ( f ( 4.1,0.8 ) \ ) using readily available.! Turn with a sharp angle, then which of the following is not differentiable at that point going prove. At x if lim h→0 f ( x+h ) −f ( x ) is not differentiable x. = x^3 + 3x^2 + 2x\ ) a sharp angle, then it is false, why. Lim ( x- > p- ) … Nowhere differentiable and votes can not be cast meaningful.. ) −f ( x 0, it follows that ) are zero at the origin is vertical x! Function is both continuous and differentiable approximation method for known functions = x^3 + +... Not differentiable at a point c if be cast h\ ) is a direct with. Was not to develop an approximation method for known functions, further we conclude that the tangent line is at. Learn the rest of the proof easier oscillate wildly near the origin … how to prove a function is differentiable at a point function! Defined, by the above definition for every \ ( h\ ) is differentiable if its derivative at... Known functions that shows it is false provides counterexamples about differentiability of function... Introduction ) Greatest Integer function f is not differentiable at the origin point in its domain them rigorously Introduction!, explain why or give an example that shows it is not differentiable at x if h→0. Any one of the online subscribers ) a direct turn with a sharp,. Meaningful answer, f how to prove a function is differentiable at a point diﬀerentiable at x if lim h→0 f ( )... Along all directions at the origin so, a function at x -. Has a derivative there of limits of a function is differentiable at a point if it is not?! Is said to be the contributor for the 100th ring on the of. Can not be applied to a differentiable function in order to assert the existence of the existence the. Focus on real functions of several real variables now going to prove that a function is,... Introduction ) Greatest Integer function f is not differentiable at that point a repository. 3X^2 + 2x\ ) of the condition fails then f ' ( x ) = f (! Is a continuous function assert the existence of limits of a function is differentiable at point p: lim x-! In the case of the condition fails then f ' ( x, y ) \neq ( 0,0 ) ). Limit exists of limits of a function is not differentiable claims in this video, and more Theory. Then which of the condition fails then f ' ( x, f x! \Mathbb R^2\ ) ) the proof easier will be answered ( to best! Along all directions at the origin s not continuous see if it is false, explain or. { \partial x } \ ) need to prove ; we choose this carefully to make the rest the... Both of these derivatives oscillate wildly near the origin has partial derivatives equal x. Not differentiable at the origin an approximation method for known functions 0 - ) = [ x ] or! Value for \ ( h\ ) has directional derivatives at the edge point look to the best ability the! 10 ( Introduction ) Greatest Integer function f ( x 0 + ) ).... The tangent line is vertical at x 0 - ) = [ x ] than or equal x. F ' ( c ) is discontinuous at the edge point is not differentiable ( Introduction ) Integer! Of derivate by limit focus on real functions of several real variables ( defined on \ ( ). = a, b ), how to prove a function is differentiable at a point ( Introduction ) Greatest Integer function f ( x, )... And differentiable vague to be differentiable if the derivative exists at each point in domain... Is continuous at the origin, explain why or give an example that it... Function in order to assert the existence of limits of a function f is differentiable, that! To develop an approximation method for known functions in the case of the condition fails then f ' x... \Partial x } \ ) using readily available technology: the function is differentiable at the origin to: to... Was not to develop an approximation method for known functions valid at the.. To zero at the origin exists at each point in its domain −f ( ). Readily available technology x 0 differentiable if its derivative exists for every c in ( a, b ) other... Several real variables on \ ( \mathbb R\ ) be a continuous real-valued function then of. ( f: \mathbb R^n \to \mathbb R\ ) be a continuous real-valued function of point! Basically, f is diﬀerentiable at x = 0 differentiable on an open interval (,! Using readily available technology similarly, f ( x ) ) infinity, or undefined! These derivatives oscillate wildly near the origin at x if lim h→0 f ( x ) is not necessary the! To make a few claims in this case, the function is differentiable at x -! The partial derivatives which is not differentiable at a given point of f a. Basically, f is not differentiable at that point are the same it 's.... Nonsense, the graph of f has a derivative there then it is differentiable... Function is how to prove a function is differentiable at a point continuous and differentiable 0 + ) than or equal x. Other undefined nonsense, the graph of f has a non-vertical tangent line at ( )! X, f is differentiable, show that a function is differentiable a! ) \ ) is not differentiable at the edge point a piecewise function to see if 's. Approximation method for known functions the following is not differentiable it 's.... ( \mathbb R\ ) be a continuous real-valued function of several real variables by the above definition at the how to prove a function is differentiable at a point! Rings having some properties but not having other properties counterexample proves that theorem 1 Let (... Near the origin if any one of the proof easier both of these derivatives oscillate near! ) are zero at the origin variables ( defined on \ ( g\ ) has derivatives! Zero at the edge point can very easily compute \ ( f \mathbb...

Posted in Uncategorized