Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-30
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  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Extension by zero without support away from the boundary is not smooth

Statement refuted

Any smooth function on an open set extends smoothly to the ambient manifold by setting it equal to zero outside the open set.

Facts & Assumptions

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

[L1]

Smooth extension works only after the support is kept away from the boundary by a cutoff (Smooth extension from a closed neighbourhood).

Counterexample

technique · direct
1.1

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

given
2.1

The function F is not continuous at 0, so it is not smooth.

step 1.1
3.1

Hence the hypothesis singled out in [L1] is essential.

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