What's an example of a function that is continuous at $0$ but is not second-differentable at $0$?

Question

What's an example of a function $f:\mathbb R\to\mathbb R$ where $f$ is continuous at $0$ but is not second-differentable at $0$?

Answer 1

Take$$f(x)=\begin{cases}x^2&\text{ if }x\geqslant0\\-x^2&\text{ otherwise.}\end{cases}$$

Answer 2

$f(x)=|x|$ is continuous at $0$ but is not differentable at $0$ so not second-differentable at $0$

Answer 3

I suppose you mean 1 time differentiable at 0, then let consider

$$f(x)=\begin{cases}x^2&\text{ if }x\geqslant0\\0&\text{ otherwise}\end{cases}$$

thus

$$f’(x)=\begin{cases}2x&\text{ if }x\geqslant0\\0&\text{ otherwise}\end{cases}$$

and $f’’(0)$ is not defined.

Answer 4

Let $$ f(x)=\begin{cases}x^2&x\in\Bbb Q\\0&x\notin \Bbb Q\end{cases}$$

Then $f$ is continuous at $0$ (but nowhere else), and is differentiable at $0$ (with $f'(0)=0$) but nowhere else; consequently, $f$ is nowhere twice differentiable.