Suppose that a ∊ R, that £ is a nonempty subset of R, and that f, g : E → R are uniformly continuous on E.
a) Prove that f + g and αf are uniformly continuous on E.
b) Suppose that f, g are bounded on E. Prove that fg is uniformly continuous on E.
c) Show that there exist functions f,g uniformly continuous on R such that fg is not uniformly continuous on R.
d) Suppose that f is bounded on E and that there is a positive constant ε0 such that g(x) > ε0 for all x ∊ E. Prove that f/g is uniformly continuous on E.
e) Show that there exist functions f,g, uniformly continuous on the interval (0,1), with g(x) > 0 for all x ∊ (0, 1), such that f/g is not uniformly continuous on (0,1).