example of differentiable function which is not continuously differentiable

Let $f$ be defined in the following way:

f(x)=\begin{cases}x^{2}\sin\left(\frac{1}{x}\right)&\text{if }x\neq 0\\ 0&\text{if }x=0.\end{cases}

Then if $x\neq 0$ , $f^{\prime}(x)=2x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x}\right)$ using the usual rules for calculating derivatives. If $x=0$ , we must compute the derivative by evaluating the limit

\lim_{\epsilon\to 0}\frac{f(\epsilon)-f(0)}{\epsilon}

which we can simplify to

\lim_{\epsilon\to 0}\,\epsilon\sin\left(\frac{1}{\epsilon}\right).

We know $\left|\sin(x)\right|\leq 1$ for every $x$ , so this limit is just $0$ . Combining this with our previous calculation, we see that

f^{\prime}(x)=\begin{cases}2x\sin\left(\frac{1}{x}\right)-\cos\left(\frac{1}{x% }\right)&\text{if }x\neq 0\\ 0&\text{if }x=0.\end{cases}

This is just a slightly modified version of the topologist’s sine curve; in particular,

\lim_{x\to 0}f^{\prime}(x)

diverges, so that $f^{\prime}(x)$ is not continuous, even though it is defined for every real number. Put another way, $f$ is differentiable but not $C^{1}$ .

Title	example of differentiable function which is not continuously differentiable
Canonical name	ExampleOfDifferentiableFunctionWhichIsNotContinuouslyDifferentiable
Date of creation	2013-03-22 14:10:18
Last modified on	2013-03-22 14:10:18
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	8
Author	Koro (127)
Entry type	Example
Classification	msc 57R35
Classification	msc 26A24