6. Suppose that 0: E R, E is a nonempty subset of R, and j, g : E ~ Rare uniformly continuous on E.

(a) Prove that j + g and o:j are uniformly continuous on E.

(b) Suppose that j, g are bounded on E. Prove that j g is uniformly continuous on E.

(c) Show that there exist functions j, g uniformly continuous on R such that j g is not uniformly continuous on R.

(d) Suppose that j is bounded on E and that there is a positive constant EO such that g(x) ~ EO for all x E E. Prove that j jg is uniformly continuous on E.

(e) Show that there exist functions j, g, uniformly continuous on the interval

(0,1), with g(x) > ° for all x E (0,1), such that j jg is not uniformly continuous on (0,1).

(f) Prove that if j, g are uniformly continuous on an interval [a, b] and g(x) i= °

for x E [a, b], then j j g is uniformly continuous on [a, b].