PropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-29

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Smooth maps are continuous

Statement

Let $M$ and $N$ be smooth manifolds and let $F : M \to N$ . If $F$ is of class $C^{r}$ at $p$ for some $r \geq 1$ , then $F$ is continuous at $p$ ; the same holds when $F$ is smooth at $p$ . Consequently every map that is $C^{r}$ (or smooth) on an open set is continuous on that open set. For $r = 0$ continuity is part of the definition and is asserted, not proved.

Facts & Assumptions

Given: Smooth manifolds $M, N$ , a map $F : M \to N$ , a point $p$ , and $r \geq 1$ such that $F$ is $C^{r}$ at $p$ .

[F1]

$C^{r}$ at $p$ means that for smooth charts $(U, φ)$ at $p$ and $(V, ψ)$ at $F (p)$ with $F (U) \subseteq V$ , the representative $ψ \circ F \circ φ^{- 1}$ is $C^{r}$ near $φ (p)$ ; for finite $r$ every iterated coordinate partial derivative of order at most $r$ exists and is continuous ( $C^{r}$ and smooth maps between smooth manifolds, $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[F2]

Charts are homeomorphisms (Manifold charts, coordinate domains, and coordinate functions).

[L1]

A map with continuous first partial derivatives is totally differentiable and therefore continuous (If all partial derivatives exist on a neighbourhood and are continuous at a point, then the map is totally differentiable there with Jacobian derivative, Total differentiability gives a local $O (∥ h ∥_{2})$ increment bound and therefore continuity).

[L2]

Continuity is local on the source, and composites of continuous maps are continuous (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

Proof

technique · direct

1.1

Choose smooth charts $(U, φ)$ at $p$ and $(V, ψ)$ at $F (p)$ with [given, F1, L1, choose] $F (U) \subseteq V$ . Since $r \geq 1$ , [F1] gives the representative $ψ \circ F \circ φ^{- 1}$ of class $C^{r}$ , hence $C^{1}$ , near $φ (p)$ , so its first partial derivatives exist and are continuous there; [L1] makes it continuous on a neighbourhood of $φ (p)$ .

givenF1L1choose

2.1

On $U \cap F^{- 1} (V)$ the map $F$ equals [given, F2, L2, step 1.1] $ψ^{- 1} \circ (ψ \circ F \circ φ^{- 1}) \circ φ$ : the three factors are continuous, $φ$ and $ψ^{- 1}$ because [F2] makes charts homeomorphisms and the middle factor by step 1.1, so [L2] makes the composite continuous. The set $U \cap F^{- 1} (V)$ contains a neighbourhood of $p$ and the agreement holds there, so by the locality clause of [L2] the map $F$ is continuous at $p$ .

givenF2L2step 1.1

3.1

The smooth case is the case $r = \infty$ , which includes $r = 1$ ; the assertion [given, step 2.1] on an open set follows by applying the pointwise statement at every point. For $r = 0$ the definition already requires continuity.

givenstep 2.1∎

Depends on

Used by

Diffeomorphisms and local diffeomorphisms of manifolds Definition
Smooth maps paste over an open cover Lemma
Identity maps and composites of smooth maps are smooth Proposition

Dependency tree · two levels

30 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Nigel Hitchin, Differentiable Manifolds, §2.4, Exercise 2.3 (standard reference, not scraped)
Rob van der Vorst, Introduction to differentiable manifolds, §2, Theorem 2.15 (standard reference, not scraped)