Lebl-Basic-Analysis-I · Problem 3.2.19

Continuous Functions

Mathematics · continuous-functions

Problem

(Challenging) A function

f : I \to \mathbb{R}

is convex if whenever

a \leq x \leq b

for

a, x, b

I

, we have

f(x) \leq f(a)\dfrac{b-x}{b-a} + f(b)\dfrac{x-a}{b-a}

. In other words, if the line drawn between

\bigl(a, f(a)\bigr)

and

\bigl(b, f(b)\bigr)

is above the graph of

f

. a) Prove that if

I = (\alpha, \beta)

an open interval and

f : I \to \mathbb{R}

is convex, then

f

is continuous. b) Find an example of a convex

f : [0, 1] \to \mathbb{R}

that is not continuous.