I assume the three conditions should be formulated for g and not f. Obviously, g(0) = 0. For the other two conditions, it is easier to apply the ε-δ definition of continuity. Suppose you are given an ε > 0. Which δ > 0 can you choose so that |x - 0| < δ would guarantee that |g(x) - g(0)| < ε? Note that |g(x) - g(0)| = |x * f(x)| = |x| * |f(x)| and |f(x)| ≤ 10.

You can only apply the product rule when both functions are differentiable at the given point. Here f(x) is not even continuous at 0. Try using the definition of derivative as a limit.