CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passaudited 2026-07-31

How 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.

Every real-analytic function is infinitely differentiable

Statement

Every real-analytic function on an open subset $U \subseteq R$ is infinitely differentiable at every point of $U$ .

Facts & Assumptions

Given: A real-analytic function $f : U \to R$ .

[L1]

Near every $c \in U$ , $f$ equals a convergent power series centred at $c$ (A real-analytic function on an open subset of $R$ is locally represented by a convergent real power series).

[L2]

A power-series sum is infinitely differentiable inside its radius (A power-series sum is infinitely differentiable inside its radius and satisfies $a_{n} = f^{(n)} (c) / ι (n!)$ at its centre).

Proof

technique · direct

1.1

Fix $c \in U$ and choose the local representing power series supplied by [L1].

givenL1choose

2.1

By [L2], that series, hence $f$ , has derivatives of every order on a neighbourhood of $c$ . Since $c$ was arbitrary, $f$ is smooth on $U$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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

Analytic function, Encyclopedia of Mathematics (standard reference, not scraped)
E. Randles, Supplementary Notes for Real Analysis (standard reference, not scraped)