If f,g cont. , then "g o f" cont.? 1. The problem statement, all variables and given/known data Let X,Y,Z be spaces and f:X-->Y and g:Y-->Z be functions. If f,g are continuous, then so is g o f. 2. Relevant equations 3. The attempt at a solution This is my proof so far. I would be appreciative if someone could point out the weak points. "Proof". Let f,g be continuous. Show "g o f" continuous. Show g(f(x)) continuous. Since f is continuous, and since f:X-->Y, we know that x is an element of W such that f[W][tex]\subseteq[/tex]V for V open, V[tex]\subseteq[/tex]Y. Since g is continuous, and since g:Y-->Z, we know that f(x) is an element of V such that g[V][tex]\subseteq[/tex]M, for M open, M[tex]\subseteq[/tex]Z. Therefore, g[ f(x) ] [tex]\in[/tex] M [tex]\subseteq[/tex] Z. Thus, "g o f" is continuous.