example of differentiable function which is not continuously differentiable

Let $f$ be defined in the following way:

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

$$\underset{ϵ\to 0}{lim}\frac{f(ϵ)-f(0)}{ϵ}$$

which we can simplify to

$$\underset{ϵ\to 0}{lim}ϵ\sin \left(\frac{1}{ϵ}\right).$$

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

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

$$\underset{x\to 0}{lim}{f}^{\prime }(x)$$

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 $C^1$.