# Open interval or Closed interval in defining convex function

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• junyoung
In summary, the conversation discusses the definition of convexity in mathematics, specifically in the context of functions and intervals. While some textbooks define convexity as an open section, others define it as a closed section. The definition of convex function is also mentioned, and there is a discussion about whether it is appropriate to define convexity for open, closed, or half-open intervals. Ultimately, the general consensus is that it is reasonable to define this property on all types of intervals.f

#### junyoung

TL;DR Summary
Open interval or Closed interval in defining convex function. I think it is proper not to include the end point because of its definition. Your opinion??
The Korean textbook standard defines the convexity of the function as an open section. Many textbooks and university calculus textbooks define the convexity of the curve as an open section. However, some textbooks define convexity as closed sections.
Do you think it is right to define the convexity for the open section or for the closed section?

The definition of convex function: Consider a function y = f (x), which is assumed to be continuous on the interval [a, b]. The function y = f (x) is called convex downward (or concave upward) if for any two points x1 and x2 in [a, b], the following inequality holds: f{(x1+x2)/2}≤{f(x1)+f(x2)}2. If this inequality is strict for any x1, x2 ∈ [a, b], such that x1 ≠ x2, then the function f (x) is called strictly convex downward on the interval [a, b].

I think open interval is more close to answer. Because the definition of convex function is ‘a function f(x) is said to be convex at an interval [a,b] for all pairs of points on the f(x) graph, the line segment that connects these two end points passes above f(x) the curve.’ Definition of word above is ‘in a higher position than something else.’(Cambridge Dictionary) So I think it is proper not to include the end point.

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• Delta2
It seems perfectly reasonable to be able to define this property on open intervals, closed intervals and half open intervals. Wikipedia at https://en.wikipedia.org/wiki/Convex_function uses the more general context of a vector space and defines it on a convex set; no reference to open or closed.

• junyoung
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