Would be enough to demonstrate that:
$f$ is continuous (1)
$∇f = 0$ at that given point - $ x^* $ (2)
$H(f) = 0 $ at that given point - $ x^* $ (3)
in order to make the case that a particular function $ f $ has only one local minimizer $ x^* $
If yes, why is this the case? What if we consider f a non-continous function then how this would change the situation?