CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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A totally differentiable map with zero derivative on a convex open set is constant

Statement

Let $U \subseteq R^{m}$ be convex and open. If $f : U \to R^{n}$ is totally differentiable at every point and $D f (z) = 0$ for every $z \in U$ , then $f$ is constant on $U$ .

Facts & Assumptions

Given: A convex open $U$ and a totally differentiable map with zero total derivative at every point.

[L1]

The total-derivative mean-value inequality implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ under a uniform derivative bound $M$ (On a convex open set, a uniform bound $∥ D f (z) v ∥_{2} \leq M ∥ v ∥_{2}$ implies $∥ f (y) - f (x) ∥_{2} \leq M ∥ y - x ∥_{2}$ ).

Proof

technique · direct

1.1

The zero derivative hypothesis satisfies the bound in [L1] with $M = 0$ for arbitrary $x, y \in U$ .

2.1

Hence $∥ f (y) - f (x) ∥_{2} \leq 0$ , so norm separation gives $f (y) = f (x)$ .

step 1.1algebra

3.1

Since $x, y$ were arbitrary, the map is constant; if $U$ is empty this conclusion is vacuous.

step 1.1step 2.1∎

Depends on

On a convex open set, a uniform bound $\|Df(z)v\|_2\le M\|v\|_2$ implies $\|f(y)-f(x)\|_2\le M\|y-x\|_2$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

J. Lebl, Basic Analysis I, §8.4 (standard reference, not scraped)