Finding where a function is increasing or decreasing: regions to check

Question

I'm solving a problem where I need to determine where a function is increasing and decreasing. I know that I need to check the regions between the critical points. However, the proposed solution to the problem also says that I need to include points where the function itself is undefined. This makes sense to me in the case of a function like $\frac{1}{x}$: it could change from increasing to decreasing around $x = 0$. But I cannot figure out how this generalizes. If the function is defined on a closed interval, then it is defined at every point on the interval. It may have domain $[a,b] \setminus \{0\}$, which would cause me to check $x = 0$, but what about $x < a$ or $x > b$? Should I, by default, check every real number for which the function is not defined?