(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Assume f is a continuous real function defined in (a,b) such that f([tex]\frac{x+y}{2}[/tex])<=[tex]\frac{f(x)+f(y)}{2}[/tex] for all x,y in(a,b) then f is convex.

2. Relevant equations

3. The attempt at a solution

my attempt is to suppose there are 3 points p<r<q such that f(r)>g(r), where g(x) is the straight line connecting (p,f(p)) and (q,f(q)), and get a contradiction.

1, There exists a neighborhood of r such that any x in N(r) implies that f(x)>g(x), followed by continuity.

2, Let E_0={p,q}, define E_1 = E_0[tex]\cup[/tex]{x | x is a mid point of any two points in E_0}. we continue this procedure and let E= union of all E_n.

3, For any points x in E, f(x)<=g(x), which can be proved straightforwardly

4, then we can find a point t in E which is also in N(r) such that f(x)<=g(x), if E is proved to be dense in [p,q].

5, we get a contradiction.

Question 1, I got stuck when I came to prove that E is dense in [p,q], it is jammed...Could anyone give me some hint to prove E is dense?

Question 2, Is there any other way(easier or harder are both welcome) to prove it?

Question 3, by the way, how can I determine whether a point belongs to Cantor set or not? Say, 1/4?1/5?

Thanks a lot!

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# Homework Help: Convexity of continuous real function, midpoint convex

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