Let f A R and g A R be functions and dene h A R to be the

Let f : A R and g : A R be functions and dene h : A R to be their product: h(x) = f(x)g(x). If h is continuous on A, then f and g are continuous on A. Is this true or false? Prove or disprove your answer.

Solution

FALSE:

Let A =[0,1] and define

f(x) =1 if x is rational

      =0 otherwise

g(x) = 1 if x irrational

         =0 otherwise.

Then h(x) is identically zero on A , hence continuous.

But neither f nor g is continuous.