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**If f,g cont. , then "g o f" cont.?**

## Homework Statement

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.

## Homework Equations

## 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.