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

$\blacksquare$