Let $f$ be real function which is differentiable everywhere except at a finite set of singularity point $S$.
Assume that $\lim_{x\to \alpha} f(x) = -\infty$ for every $\alpha \in S$. Is it possible to prove that the global maxima is a local maxima? (i.e it can be found with the first and second derivative test?)
I'm not very strong in analysis so any help would be appreciated, thanks.
No. After the last singularity point, the function could increase so $\lim_{x\to\infty}f(x)=\infty$. Here, there would be no global maximum. If a global maximum exists, it occurs at a local maximum since this is true of all differentiable functions.