PropositionStatement: Literature-sourcedProof: AI-generatedverified 2026-09-26 (gpt-6-sol)

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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]

The published definition requires $F$ to be continuous at $p$ before calling it $C^{r}$ at $p$ . It also requires smooth charts $(U, φ)$ at $p$ and $(V, ψ)$ at $F (p)$ with $F (U) \subseteq V$ for which the representative $ψ \circ F \circ φ^{- 1}$ is $C^{r}$ near $φ (p)$ ( $C^{r}$ 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

1.1F1

Continuity at $p$ follows immediately from the definition in [F1]. The same definition requires continuity on an open set when a map is called $C^{r}$ there. Thus the first and open-set conclusions hold directly, including the $r = 0$ case.

1.2F1F2L1L2

The coordinate condition gives the same conclusion directly when $r \geq 1$ : on a neighbourhood of $φ (p)$ , the representative has continuous first partial derivatives and is continuous by [L1]. There $F = ψ^{- 1} \circ (ψ \circ F \circ φ^{- 1}) \circ φ$ . Charts and their inverses are continuous by [F2], so this composite is continuous near $p$ . The required chart neighbourhood with $F (U) \subseteq V$ is already part of [F1].

2.1F1step 1.1step 1.2∎

A smooth map is $C^{1}$ , and continuity on an open set also follows pointwise. The chart calculation in step 1.2 is compatible with, but not needed for, the continuity premise built into the published definition.

Depends on

Used by

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)