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?
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1If the domain of your function is a union of intervals (some may have an infinite end), you only need to include in your study the ends of all these intervals. – Anne Bauval Dec 28 '22 at 08:24
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@AnneBauval As a quick follow-up on that, let's say that I were considering $f(x) = 1/x$ defined on $\mathbb{R} \setminus {0}$. It would be natural to include $x = 0$ in that case, right? (Some authors seem to call this "including points of discontinuity," but since $f$ isn't defined at $x = 0$, I don't think I can say $f$ is discontinuous at $x = 0$.) – Mathematical Endeavors Dec 28 '22 at 08:33
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Right. Including such points is not for the study of continuity but for that of monotonicity. – Anne Bauval Dec 28 '22 at 08:39