Question

If f and g are uniformly continuous anf bounded then fg is uniformly continuous

Answer 1

Let f and g be uniformly continuous on A. Then given ϵ>0 there exists a δ1>0 such that if |x−y|<δ1,∀x,y∈A, then |f(x)−f(y)|<ϵ2M. There also exists a δ2>0 such that if |x−y|<δ2,∀x,y∈A, then |g(x)−g(y)|<ϵ2M. Since f and g are both bounded on a, there exists M1>0 such that |f(x)|≤M1,∀x,y∈A and there exists M2>0 such that |g(x)|≤M2,∀x,y∈A. Let M={M1,M2}. Then |f(x)|≤M and |g(x)<M for all x∈A. Let δ={δ1,δ2}. So if |x−y|<δ then |f(x)−f(y)|<ϵ2M and |g(x)−g(y)|<ϵ2M. Now consider |f(x)g(x)−f(y)g(y)|=|f(x)g(x)−g(x)f(y)+g(x)f(y)−f(y)g(y)|. Then, |g(x)||f(x)−f(y)|+|f(y)||g(x)−g(y)|≤M|f(x)−f(y)|+M|g(x)−g(y)|<Mϵ2M+Mϵ2M=ϵ. Do