Let X Y Z be metric spaces Suppose f X rightarrow Y and g Y

Let X, Y, Z be metric spaces. Suppose f: X rightarrow Y and g: Y rightarrow Z are uniformly continuous functions. Show that g o f: X rightarrow Z is uniformly continuous.

Solution

Given f and g are continuous function.

Let f is continuous at x₀ X and g is continuous at f(x₀) Y

Let e > 0 be any arbitrary small number.

Since g is continuous at f(x₀), we can choose ₀ > 0 such that

if d_Y (y, f(x₀)) < ₀ then d_Z(g(y), g(f(x₀))) < e.

Since f is continuous at x₀, we can choose > 0 such that

if d_X(x, x₀) < then d_Y (f(x), f(x₀)) < ₀ .

Putting these together, if d_X(x, x₀) < then d_Y (f(x), f(x₀)) < ₀ ,

giving d_Z(g(f(x)), g(f(x₀))) < e.

This means d_Z((g f)(x),(g f)( x₀)) < e and g f is continuous at x₀