I'll be very thankful is someone will tell me where I'm wrong. We know: 1) f is uniform continuous. 2) g is uniform continuous. We want to prove: fg(x) is uniform continuous. proof: from 1 we know -> for every |a-b|<d_0 exists |f(a)-f(b)|<e from 2 we know -> for every |x-y|<d exists |g(x)-g(y)|<d_0 let a=g(x) and b=g(y) then for every x,y |x-y|<d exists |fg(x)-fg(y)|<e. Sorry for my poor formulation, English is not my mother tongue.