Convexity and Continuity

Not to disappoint I shall at least make one new post during my winter break. However, do not look for this to be a recap of how things went. If I could I would rather just forget that these three weeks or so ever happened. On the bright side I did however, manage to get some productive things done. I finished reading the chapter on continuity and have almost finished reading the chapter on differentiation. While working through the continuity exercised I ran into one that was particularly interesting problem. Before going any further I would like to mention that I had to ask for help in finding this solution.

Theorem:
Every real-valued convex function is continuous.

A real-valued function

defined in is said to be convex if



whenever and .

Lemma: If is continuous in (a,b) and if then



Proof of lemma: This is not to difficult to prove so I shall just give the major idea and leave the rest to the reader. Let . Obviously , so use this new way to express and the rest follows with ease.

Proof of theorem:

First we choose a point

and a neighborhood of such that . Now using the lemma we have that

, for .

Now from this we see that there is a positive number such that . Let . Thus if we have that .