Global maximum for a two times continuously differentiable function

Question

Let $M:=\{(x,y)\in \mathbb R^2:x^2+y^2<1\}\setminus\{(x,0)\in \mathbb R^2, x\in \mathbb R\}$ and $f:\mathbb R^2\rightarrow \mathbb R$ be two times continuously differentiable.

I want to know whether the restriction $f:\overline{M}\rightarrow \mathbb R$ of $f$ attains its global maximum on $\overline{M}$.