Question: (1). Let f and g be continuous on . Prove that if if f(a)g(a) for some a, then there exists a >0 such that f(x)g(x)

(1). Let f and g be continuous on . Prove that if if f(a)g(a) for some a, then there exists a >0 such that f(x)g(x) whenever |x-a|<. (2). Let f be continuous on . Prove that if f(x) is rational for every x then f is a constant function. I think I need to use reduction to the absurd to prove (1) and (2) but I don't know how to prove these problems

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