Alphabeta MathHow statement and proof provenance work
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.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
has both partial derivatives at the origin but is discontinuous there
Statement refuted
If both partial derivatives of a real function exist at a point, then the function is continuous there.
Facts & Assumptions
Given: The function and when .
Partial derivatives are directional derivatives in the standard basis directions (Directional derivatives and partial derivatives of a map ).
A map is continuous at if its values tend to as the input tends to (Continuity of a map between metric spaces, at a point and globally, in the - form).
Counterexample
Both axis restrictions of are identically zero, so [L1] gives .
On the punctured diagonal , .
As but the values in step 2.1 do not tend to , [L2] shows that is discontinuous at the origin.
Depends on
- Directional derivatives and partial derivatives of a map $U\subseteq\mathbb{R}^m\to\mathbb{R}^n$
- Vector-valued functions $f : A \to \mathbb{R}^m$, their limits and continuity, with the dictionary to the metric notions
- Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
22 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, §8.3 (standard reference, not scraped)