Composition of continuous functions is continuous

Builds on Continuous function Epsilon-delta definition of continuity

Statement

Let $D, E \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ be a function with $f(D) \subseteq E$ , and let $g \colon E \to \mathbb{R}$ . If $f$ is continuous at $a \in D$ and $g$ is continuous at $f(a)$ , then $g \circ f$ is continuous at $a$ . In particular, if $f$ and $g$ are continuous, then $g \circ f$ is continuous.

Remarks

Given $\varepsilon > 0$ , choose $\eta > 0$ for $g$ at $f(a)$ , and then $\delta > 0$ for $f$ at $a$ and the tolerance $\eta$ . See the epsilon-delta form. For example, $x \mapsto \sin(x^2)$ is continuous. Mathlib states it for topological spaces, as Continuous.comp.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Continuous function definition
Epsilon-delta definition of continuity statement
Composition of continuous functions is continuous statement

Tag	`0NRQPQX3VA`
URI	`m2://0NRQPQX3VA`
Kind	statement, proposition
Writing	W0 stub
MSC	26A15
Status	This copy of the site has no build data. Run `m2 status 0NRQPQX3VA`.
Uses	Continuous function, Epsilon-delta definition of continuity

Composition of continuous functions is continuous

Statement

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by