Is local minimum/maximum necessarily global when it's the only stationary point of a continuous & differentiable function?

Question

Couldn't find this theorem even though it feels very intuitive to me.

If the $f:R^n \to R$ is continuous, and has only one stationary point - a local minimum/maximuma. Doesn't it necessarily makes it global?

If not - can you please give an example?

If yes - where is it proven?

Answer 1

For $n=1$: you need $f \in C^1$ (the function is not only continuous but continuously differentiable). (a counterxample: $f(x) = e^x - |x + 1|$)

For $n>1$ things are more complicated. See here