See theorem 3.2 here W.rudin, Real and Complex Analysis . Ask your doubts.
I need to prove that a real valued convex function on R is continuous. I'm thinking this is only a special case and may not be the right way to attack the problem:
Assume the contrary. That is, there is at least one point such that for any there is no such that implies |f(x_0) - f(x)| < \epsilon.
Now, if we further assume that there exists such that and d(f(x),f(y))<\epsilon, then it follows that since x_0 = \lambda x + (1-\lambda) y for some we have
|f(x_0) - f(x)| = |f(\lambda x + (1-\lambda) y) - f(x)| \le (1-\lambda)|f(y)-f(x)| \le \epsilon
This seems like an elegant application of convexity, but it only applies to simple discontinuities, and I'm not sure I can extend it to the more general case.
Thanks!! edit: not sure whats wrong with my latex :/