Question

I need to prove continuity proposition.

Answer 1

if f is continuous at z0 in S and g is continuous at f(z0) then gf is continuous at z0

Answer 2

is continuous at z

Answer 3

The step to proof is really one of comprehension: just note that when you compose two functions you treat one (in this case f) as the x or z value value of the other, and you've already established that g is continuous at f of z sub zero, the point in question. Write a paragraph proof maybe.