Limits and Continuity, Composing Functions and Limits
Composing Functions and Limits
The composition of continuous functions is continuous.
Pull ε back to δ, back to γ,
and when x is within γ of p,
f(x) is within δ of f(p),
and g(f(x)) is within ε of g(f(p)).
The same proof shows the composition of uniform functions is uniform.