False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30

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A continuous partition of unity need not be smooth

Statement

False claim: every continuous partition of unity on a smooth manifold is smooth.

Facts & Assumptions

Given: The cover $U = (- \infty, 1)$ and $V = (- 1, \infty)$ of $R$ .

[F1]

A partition of unity subordinate to an open cover consists of nonnegative functions summing to $1$ with supports in the assigned open sets (Smooth partitions of unity subordinate to an open cover).

[A1]

Define $ϕ (x) : = 0$ for $x \leq - 1$ , $ϕ (x) : = (x + 1) / 2$ for $- 1 \leq x \leq 1$ , $ϕ (x) : = 1$ for $x \geq 1$ , and $ψ : = 1 - ϕ$ .

Refutation

technique · direct

1.1

The functions from [A1] are continuous, nonnegative, and satisfy $ϕ + ψ = 1$ ; also $supp (ϕ) = [- 1, \infty) \subseteq V$ and $supp (ψ) = (- \infty, 1] \subseteq U$ , so they form a continuous subordinate partition in the sense of [F1].

A1F1

1.2

The function $ϕ$ has a corner at $- 1$ and at $1$ , so it is not smooth.

2.1

Hence a continuous partition of unity need not be smooth.

step 1.1step 1.2∎

Depends on

Smooth partitions of unity subordinate to an open cover

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)