False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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Continuous dependence does not by itself imply differentiable dependence

Statement

False claim: once solutions depend continuously on data, they automatically depend differentiably on that data.

Facts & Assumptions

Given: The parameter-dependent scalar ODE $x^{'} (t) = ∣ λ ∣$ , $x (0) = 0$ .

[L1]

Continuous dependence on initial data and parameters is weaker than the $C^{1}$ and smooth dependence theorems that require derivative hypotheses (Continuous dependence of ODE solutions on initial data and parameters, $C^{1}$ dependence of solutions on initial data, Smooth dependence of solutions on initial data).

Refutation

technique · direct

1.1

For each parameter $λ \in R$ , the unique solution is [L1] $x (t; λ) = ∣ λ ∣ t$ . This depends continuously on $λ$ for every fixed $t$ .

2.1

At every fixed $t > 0$ , the map $λ \mapsto ∣ λ ∣ t$ is not [step 1.1] differentiable at $λ = 0$ . Thus continuous dependence does occur, but differentiable dependence fails.

step 1.1

3.1

Therefore continuous dependence alone does not imply differentiable [L1, step 2.1] dependence.

L1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

11 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, Appendix §10.3 (standard reference, not scraped)
Chin-Lung Wang, Banach Calculus, §4.4 (standard reference, not scraped)