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

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.

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.