If x* is a local optimum of a function in all directions, could it be optimal in a neighborhood of x*?

Question

In a Euclidean space whose base is denoted $(e_1,e_2,...e_N)$. suppose that $x*$ is the local minimum of a function $f$ in all direction. Could we say that $x*$ is an optimum of $f$ in a neighborhood of $x*$ ??

Answer 1

Thats what a local minimum means. f has a local extremum at x$^*$ if for some $\epsilon$-neighborhood of x$^*$, f(x) $\ge$ f(x$^*$) for all x in the neighborhood.