Limit of Composite Function/Corollary

Corollary to Limit of Composite Function
Let $I$ and $J$ be real intervals.

Let:
 * $g: I \to J$ be a real function which is continuous on $I$
 * $f: J \to \R$ be a real function which is continuous on $J$.

Then the composite function $f \circ g$ is continuous on $I$.

Proof
This follows directly and trivially from:
 * the definitions of continuity at a point
 * continuity on an interval
 * Limit of Composite Function.