1. Find two real-valued functions f and g that are uniformly continuous on a set D, but such that their product f g is not uniformly continuous on D. 2. Let f : D →  be uniformly continuous on the...

1. Find two real-valued functions f and g that are uniformly continuous on a set D, but such that their product f g is not uniformly continuous on D.

2. Let f : D →
 be uniformly continuous on the bounded set D. Prove that f is bounded on D.

3. (a) Let f and g be real-valued functions that are bounded and uniformly continuous on D. Prove that their product f g is uniformly continuous on D.

(b) Let f and g be real-valued functions that are uniformly continuous on a bounded set D. Prove that their product f g is uniformly continuous on D.