Does empty feasible direction implies global minimum?

Question

Consider minimizing a function $f$ over the constraint set $C$. Now, if $x \in C$ with no feasible directions then x is a global minimum.

My attempt:
If from point $x$ there is no feasible direction then that implies that entire set $C$ is actually a singleton set. Hence, $x$ has to be the global minimum since it is the only point in the feasible set.

Is my reasoning correct?

Answer 1

x has to be the global minimum since it is the only point in the feasible set.