Real function/Continuous/Composition/Fact

Continuity under composition

Let ${}D\subseteq \mathbb {R}$ and ${}E\subseteq \mathbb {R}$ be subsets and let

$f\colon D\longrightarrow \mathbb {R}$

and

$g\colon E\longrightarrow \mathbb {R}$

denote functions with ${}f(D)\subseteq E$. Then the following statements hold.

1. If ${}f$ is continuous in ${}x\in D$ and ${}g$ is continuous in ${}f(x)$, then also the composition ${}g\circ f$ is continuous in ${}x$.
2. If ${}f$ and ${}g$ are continuous, so is ${}g\circ f$.

Proof, Write another proof