Composition of Computable Real-Valued Functions is Computable

Theorem
Let $D \subseteq \R^n$ and $E \subseteq \R^m$ be subsets of real cartesian space.

Let $f : D \to \R$ be computable.

Let $g_1, g_2, \dotsc, g_n : E \to \R$ be computable.

Suppose that, for every $\bsx \in E$:
 * $\tuple {\map {g_1} \bsx, \map {g_2} \bsx, \dotsc, \map {g_n} \bsx} \in D$

Then $h : E \to \R$ defined as:
 * $\map h {x_1, \dotsc, x_m} = \map f {\map {g_1} {x_1, \dotsc, x_m}, \dotsc, \map {g_n} {x_1, \dotsc, x_m}}$

is computable.

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