I was wondering about this: suppose I have a function $f: D \to \mathbb{R}$; suppose $(a, b) \subset D$ ($D$ can either be bounded or unbounded), and say $f$ is convex in $(a, b)$.
- What is the difference between writing $(a, b)$ and $[a, b]$ in terms of convexity?
I mean, if $f$ is convex in $(a, b)$ then I expect the function to exhibit a point of local/global minima, by convexity property. Though if the interval were compact, that is $[a, b]$ I could guarantee the existence of max/min via Weierstrass Theorem (independently upon convexity).
When you deal with the study of the sign of $f$ or $f'$ or $f''$, when would you include or not the extrema of the interval (provided the extrema are indeed internal point of the domain, or at least they belong to it)?
As for example in $\sqrt{x}$, $0 \in D$ but $0$ is not an internal point. When I deal with monotonicity, $\sqrt{x}$ is monotone increasing in $[0, +\infty)$. Here no ambiguity.
But let's say a function is increasing for $x\in [2, 3]$. Should I write $x \in (2, 3)$? What's the difference?
Same speech clearly holds for curvature, that is intervals of convexity and so on.