For functions on the real numbers, the definition of "differentiable" at a point is that the derivative exist at that point. So to prove that "if f and g are differentiable at x= a, then fg is differentiable there", the best thing to do is to follow the derivation of the product rule.
The derivative of fg at x= a is and now separate into two fractions. Use the fact that f and g, separately are differentiable. Also, you will need to use the fact that, since f is differentiable, f is continuous.
For (b), use the mean value theorem: for any x and y, there exist c between x and y, such that h(x)- h(y)= h'(c)(x- y). Take the absolute value of each side and use the fact that .
For (c), start with a function that is continuous but not differentiable (there is a well known example) and integrate it!