Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-30
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Naive extension by zero from an open set need not be smooth

Statement

False claim: if a smooth function is defined on an open set containing a closed set, then declaring it to be zero outside that open set always gives a smooth global extension.

Facts & Assumptions

Given: The open set (0,∞)⊆R and the smooth function f(x)=1 on it.

[L1]

Smooth extension requires a cutoff supported away from the boundary of the original open set (Smooth extension from a closed neighbourhood).

Refutation

technique · direct
1.1

The naive zero extension is F(x):=1 for x>0 and F(x):=0 for x≤0.

given
2.1

This function has a jump at 0, so it is not smooth.

step 1.1
3.1

Therefore the cutoff in [L1] is genuinely necessary.

L1step 2.1∎

Depends on

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