When the derivatives are never zero, is the maximum of a function equal to $\infty$?

Question

Consider a function $f: \mathbb{R}^M \rightarrow \mathbb{R}$ and suppose that the system of equations $$ \begin{cases} \frac{\partial}{\partial x_1}f(x)=0\\ \frac{\partial}{\partial x_2}f(x)=0\\ ...\\ \frac{\partial}{\partial x_M}f(x)=0\\ \end{cases} $$ has no solution. Can we conclude that $\max_xf(x)=\infty$? Do we need to "augment" the codomain $\mathbb{R}$ with $\infty$?

Answer 1

No. This is false even if $M=1$. Take $f(x)=\arctan x$, for instance.

Answer 2

No. Consider the function $y=arctan(x)$.