Let g : I open interval in ( → ( be convex; i.e., for every x, x' ( I and every a ( [0, 1], it holds g(ax + (1 - a)x') ( ag (x) + (1 - a) (x) + (1 - a) g (x'). Then show that
(i) g is continuous.
(ii) For every x0 ( I. there exits ( (x0) ( ( such that g(x) - g (x0) ( ( (x0) ( (x - x0), x ( I.