Give me an example of a function $f: \Bbb R \to \Bbb R$ which is differentiable at a point but not $C^1$ at that point.
I have found a function $f: \Bbb R^2 \to \Bbb R$, $f(x,y)=|xy|$ which is differentiable at $(0,0)$ but not $C^1$ at that point. But can't find one example in $\Bbb R$.
Moreover can anyone Give me an example of a function $f: \Bbb R \to \Bbb R$ which is differentiable at a point ,$C^1$ at that point but not differentiable at that point.
For the first question take $$ f\left( x \right) = \left\{ \begin{array}{l} x^2 \sin \frac{1}{x},\,\,x \ne 0 \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0 \\ \end{array} \right. $$ $f$ is differentiable at $x=0$ but the derivative not continuous at $x=0$. Moreover, $$ f'\left( x \right) = \left\{ \begin{array}{l} - \cos \frac{1}{x} + 2x\sin \frac{1}{x},\,\,x \ne 0 \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0 \\ \end{array} \right.$$
For the second take $$ f\left( x \right) = \left\{ \begin{array}{l} x^3 \sin \frac{1}{x},\,\,x \ne 0 \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0 \\ \end{array} \right. $$ $f$ is differentiable at $x=0$ and the derivative is continuous at $x=0$ but $f'$ is not differentiable at $x=0$ (not twice differentible). Moreover, $$ f'\left( x \right) = \left\{ \begin{array}{l} 3x^2 \sin \frac{1}{x} - x\cos \frac{1}{x},\,\,x \ne 0 \\ 0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x = 0 \\ \end{array} \right. $$