Composition of Computable Real Functions is Computable

Theorem
Let $f,g : \R \to \R$ be computable real functions.

Let $h : \R \to \R$ be defined as:
 * $\map h x = \map f {\map g x}$

Then $h$ is computable.

Proof
Follows immediately from:
 * Composition of Sequentially Computable Real Functions is Sequentially Computable
 * Composition of Computably Uniformly Continuous Real Functions is Computably Uniformly Continuous