Real Analysis : Bounded Continuity / Differentiability

Problem:

Let f: [0, ∞) → R be a bounded function.  For all X greater than or equal to 0, let G(x)=sup{f(t): 0 is less than or equal to t is less than or equal to x}

a) Show that if f is continuous, g is also continuous. Is the converse also true? Justify.

b) If f is differentiable and continuous, is g also differentiable and continuous? Justify.