How do I prove function convexity?

Question

I have the following task:

Prove that if $ f : I \rightarrow \mathbb{R} $ is continuous ($ I $ is a range) and $$ \forall {x,y \in I} \qquad f\left(\frac{x+y}{2}\right) \leq \frac{f(x) + f(y)}{2} $$ then $f$ is a convex function.

Can somebody give a hint what can I use to prove this? This is a homework assignment, so I'd like to try solve it myself.

Isn't that inequality the definition for a convex function? Meaning, if we know this inequality holds, doesn't that mean the function is convex just by definition? — peter.petrov, Mar 30 '15 at 07:14
@peter.petrov: The definition for convexity is that $f(tx + (1-t)y) \le tf(x) + (1-t)f(y)$ for all $x, y$ and $t\in [0,1]$. — , Mar 30 '15 at 07:17
Or this older thread: Showing that $f$ is convex given that $f(\frac{x+y}2)\le\frac{f(x)+f(y)}2$. — Martin R, Mar 30 '15 at 07:21

score 1 · Accepted Answer · answered Mar 30 '15 at 07:27

Take $x,y$ two points in $I$ and $\lambda \in (x,y)$.

You know that if $\lambda = (x+y)/2 := x'$ you're done. Otherwise, $\lambda$ is either in $(x,x')$ or in $(x',y)$.

Continue this way and you'll build two sequences converging to $\lambda$. Since $f$ is continuous, it preserves limits.

That should set you on the right path.

How do I prove function convexity?

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