continuity of convex functions, alternate proof

Let f be convex and y(a,b) be arbitrary but fixed. Then

f(λx+(1-λ)y) λf(x)+(1-λ)f(y) (1)
f(λx+(1-λ)y)-f(y) λ(f(x)-f(y))λ|f(x)-f(y)|. (2)

Fix a number c>sup{|f(u)-f(v)|:u,v(a,b)}. Then

|f(λx+(1-λ)y)-f(y)|λ|f(x)-f(y)|<λc. (3)

Given ϵ>0, let λ range over (0,ϵ/c) if ϵ/c<1, or λ=1 otherwise. Then it is easy to see that f(λx+(1-λ)y) and f(y) lie within ϵ distance of each other when λ varies as specified.

Continuity of f now follows–for x<y, the left-hand limit equals f(y) and for y<x, the right-hand limit also equals f(y), hence the limit is f(y).

Title continuity of convex functions, alternate proof
Canonical name ContinuityOfConvexFunctionsAlternateProof
Date of creation 2013-03-22 18:25:28
Last modified on 2013-03-22 18:25:28
Owner yesitis (13730)
Last modified by yesitis (13730)
Numerical id 4
Author yesitis (13730)
Entry type Proof
Classification msc 26B25
Classification msc 26A51