Local minimum implies local convexity? Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$.
It typically looks like


What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such that $f$ is convex over $(a-\epsilon,a+\epsilon)$ ?

The motivation for this question is intuition, but I can't find any valid criterion.

As pointed out by the user above you certainly need more than just continuity. You also need more than differentiability since one could consider $f(x)=x^{2}(\sin(\frac{1}{x})+1)$ for $x\neq0$ and $f(0)=0$ which has a local minimum at $0$ but $f$ is not convex in any neighbourhood of $0$. By changing the exponent of the power of $x$ you can show further that to obtain local convexity it is not enough to be just $C^{2}$ unless you have positive derivative at the point of the minimum. Even worse, you can have a $C^{\infty}$ function with a local minimum at $x=0$ but still not be convex in any neighbourhood of $0$ provided you don't assume that $f''(0)>0$. This is shown by the function $f(x)=e^{\frac{-1}{x^{2}}}(\sin(\frac{1}{x})+1)$ for $x\neq0$ and $f(0)=0$.
