Decide which of the following statements are true and which are false. Prove the true ones and provide counterexamples for the false ones.
a) If f is uniformly continuous on (0, ∞) and g is positive and bounded on (0, ∞), then fg is uniformly continuous on (0, ∞).
b) The function x log(1/x) is uniformly continuous on (0,1).
c) The function
cos x/mx + b
is uniformly continuous on (0,1) for all nonzero m. b ∊ R.
d) If f, g are uniformly continuous on an interval [a, b] and g(x) ≠ 0 for x ∊ [a, b], then f/g is uniformly continuous on [a, b].