(a) let f and g be real-valued functions that are bounded and uniformly continuo

Question

(a) let f and g be real-valued functions that are bounded and uniformly continuous on D. prove that their product fg 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 fg is uniformly continuous on D

Explanation / Answer

Let f(x) = h(x)g(x). Assume that f is continuous on D but not uniformly, then there exists e>0 such that for all d>0, there exists x and y in D such that |x-y|e. In particular, for every n>0, there exists x_n and y_n in D such that |y_n - x_n|e. Since {x_n} is a sequence in D, by the Bolzano-Weierstrass theorem, there exists a subsequence x_n_p converging to z in D. As y_n_p = x_n_p + (y_n_p - x_n_p) for every n, and {y_n_p - x_n_p} converges to 0, {y_n_p} also converges to z. Since f is continuous at z, {f(y_n_p)-f(x_n_p)} converges to f(z)-f(z) = 0. This contradicts the assumption that |f(y_n_p)-f(x_n_p)|>e for all n. Hence, f must be uniformly continuous on D.

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