- #1
boombaby
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Homework Statement
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.
Homework Equations
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!