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

Let a function f : R => R be convex. Show that f is necessarily continuous. Hence, there can be no convex functions that are not also continuous.

2. Relevant equations

3. The attempt at a solution

F is continuos if there exist [tex]\epsilon[/tex] >0 and [tex]\delta[/tex]>0 such that |x-y|< [tex]\delta[/tex] => |f(x) - f(y)| <[tex]\epsilon[/tex]

on the other hand function is convex if for some numbers a and b, with a<b an [tex]\lambda[/tex] [tex]E[/tex] (0,1) f( [tex]\lambda[/tex] a+ (1-[tex]\lambda[/tex] b) [tex]\leq[/tex] [tex]\lambda[/tex] f(a) + (1-[tex]\lambda[/tex]) f(b)

so basicaly that is what I have, I got hint that we should use this definition of continuity... but I dont have idea where to start...

help, pleease...

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# Homework Help: Proof of continuity of convex functions

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