Decide which of the following statements are true and which are false. Prove the true ones...

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 f g is uniformly continuous on (0,∞).

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b) The function x log(1/x) is uniformly continuous on (0,1).

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c) The function

is uniformly continuous on (0,1) for all nonzero m, b ∈ R.

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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].