Let f be a convex function on an open set S that is bounded above by M in a neighborhood of x0; that is, there exists an open set f containing x0 such that
f(x) ‰¥ M for every x ˆˆ U
1. Show that there exists a ball B(x0) containing x0 such that for every x ˆˆ B(x0),

2. Choose some x ˆˆ B(x0) and α ˆˆ [0, 1]. Let z = αx + (1 - α)x0. Show that x0 can be written as a convex combination of x, x0 and z as follows:

3. Deduce that f(x0) - f(z) ‰¤ α(M - f(x0)).
4. Show that this implies that f is continuous at x0.

(2x0 x)