Is this Function Convex in a neighbourhood of $a$? Suppose $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is a continuous function. Suppose that $F$ attains a local minium in a point $a$. Is true that there exists some ball centered in $a$ such that $F$ restricted to this ball is convex?

Investigate the behavior of $x^2 (\sin^2\frac1x + 1)$ at $0$.