# Limit of Composite Function/Corollary

## Corollary to Limit of Composite Function

Let $I$ and $J$ be real intervals.

Let:

$(1): \quad g: I \to J$ be a real function which is continuous on $I$
$(2): \quad f: J \to \R$ 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 definition of continuity at a point
the definition of continuity on an interval
Limit of Composite Function.

$\blacksquare$