Use the definitions of limit and continuity to show that if f[x] and g[x] are continuous at all real numbers then so is the composition function, f[g[x]].

Answer:

Proof: Let > 0 be given. Let c be a real number. We want to show that f[g[x]] = f[g[c]].
Since f[x] is continous at g[c], there is a number such that
if | t - g[c] | < , then | f[t] - f[g[c]] | < .
Now use that in the limit definition for g[x]: Since g[x] is continuous at c, there is a such that
if | x- a | < , then | g[x] - g[c] | < .
Putting stuff together, we see that if we let t = g[x], we can write:

| x- a | < ==> | g[x] - g[c] | <
==> | t - g[c] | <
==> | f[t] - f[g[c]] | <
==> | f[g[x]] - f[g[c]] | <

And so we have f[g[x]] = f[g[c]] as desired. This whole process tells us how to get from using the intermediate step .