The Composition of Continuous Functions

This page is intended to be a part of the Real Analysis section of Math Online. Similar topics can also be found in the Calculus section of the site.

The Composition of Continuous Functions

On the Properties of Continuous Functions page, we looked at some very important theorems regarding combining various functions. We saw that if $f : A \to \mathbb{R}$, and $g : A \to \mathbb{R}$ were both continuous functions at $c \in A$, and $k \in \mathbb{R}$ then:

• $f + g$ is continuous at $c \in A$.
• $f - g$ is continuous at $c \in A$.
• $k f$ is continuous at $c \in A$.
• $fg$ is continuous at $c \in A$.
• $\frac{f}{g}$ is continuous at $c \in A$ provided $g(c) \neq 0$.
• $\mid f \mid$ is continuous at $c \in A$.
• $\sqrt{ f }$ is continuous at $c \in A$ if $f(x) ≥ 0$ for $x$ near $c$.

We will now look at a theorem regarding the continuity of a compositive function, which says that for $A, B \subseteq \mathbb{R}$, and for two functions, $f : A \to \mathbb{R}$ and $g : B \to \mathbb{R}$ such that $f(A) \subseteq B$ where $f$ is continuous at $c \in A$ and $g$ is continuous at $b = f(c) \in B$, then $g \circ f : A \to \mathbb{R}$ is continuous at $c \in A$.

 Theorem 1: Let $f : A \to \mathbb{R}$ and $g : B \to \mathbb{R}$. If $f(A) \subseteq B$, and $f$ is continuous at $c \in A$ and $g$ is continuous at $b = f(c) \in B$, then $g \circ f : A \to \mathbb{R}$ is continuous at $c \in A$.