Let I be an open interval containing 0, and let f : I R be any function that is bounded on I. Define a new function g : I R by g(x) := xf(x).

(a) Prove using the e- definition of continuity that g is continuous at x = 0.

(b) Suppose a not equal to 0, a I. Prove using the e- definition of continuity that if f is continuous at x = a then g is also continuous at x = a.