Example of function being two times differentiable at a point but not $C^2$

Question

I am looking for a function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that:

• $f$ is twice differentiable at $0$
• $f''(0) > 0$
• $f''(x) < 0$ around $0$ (in particular $f''$ is not continuous).

I have seen a lot of examples based on $x^\alpha \sin(\frac{1}{x^\beta})$ but I am not sure they fit my third point.

NB: I would also be satisfied with an example with more than one variable, in which case I need the Hessian $\nabla^2 f(0)$ to be definite positive, but around $0$ the Hessian is not positive.

Answer 1

There is a theorem due to Darboux which states that the derivative of a differentiable function $g:\mathbb{R}\to\mathbb{R}$ satisfies the intermediate value property on any interval $[a,b]$. Since $f'(x)$ is differentiable in a neighborhood of $0$, $f''(x)$ satisfies the IVP on this neighborhood. Taking an interval like $[a,0]$ for negative $a$ sufficiently close to $0$ then shows such a function $f$ as you've described cannot exist.