ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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.

The one-sided flat function is $C^\infty$ with identically zero Taylor series

Example

Define $\phi(x)=\begin{cases}\exp(-1/x),&x>0,\\0,&x\le0.\end{cases}$ Then $\phi\in C^\infty(\mathbb R)$ and $\phi^{(m)}(0)=0$ for every $m\in\mathbb N$ , although $\phi(x)>0$ for $x>0$ .

Facts & Assumptions

Given: The displayed function.

[L1]

Exponential dominates every polynomial (The exponential dominates every fixed nonnegative integer power at $+\infty$ ).

[L2]

Exponential is smooth, with positive values (The exponential function is smooth and $(\exp)'=\exp$ , A power-series sum is infinitely differentiable inside its radius and satisfies $a_n=f^{(n)}(c)/\iota(n!)$ at its centre, The exponential is positive and satisfies $\exp(-x)=1/\exp(x)$ ).

[L3]

Products and composites are differentiable by Sums, scalar multiples, products and quotients: $(f+g)'(c) = f'(c) + g'(c)$ , $(\alpha f)'(c) = \alpha f'(c)$ , $(fg)'(c) = f'(c)g(c) + f(c)g'(c)$ , and $(f/g)'(c) = \bigl(f'(c)g(c) - f(c)g'(c)\bigr)/g(c)^{2}$ when $g(c) \ne 0$ and The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)'(c) = f'(g(c))\,g'(c)$ .

Verification

technique · induction

1.1

On $x>0$ , repeated product and chain rules give $\phi^{(m)}(x)=P_m(1/x)\exp(-1/x)$ for a polynomial $P_m$ ; on $x<0$ , every derivative is $0$ .

baseL2L3given

2.1

If the formula holds at order $m$ , differentiating produces another polynomial $P_{m+1}(1/x)$ times $\exp(-1/x)$ . By [L1], this tends to $0$ as $x\to0^+$ .

ihstep 1.1L1L2L3

3.1

The difference quotient for the $m$ -th derivative at $0$ is again a polynomial in $1/x$ times $\exp(-1/x)$ , so it tends to $0$ . Induction therefore extends every derivative continuously across $0$ , with value $0$ .

step 2.1L1givendischarge-induction∎

Depends on

Used by

A nonzero smooth compactly supported bump Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 104 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Analytic Functions (standard reference, not scraped)
MIT 18.102, Chapter 4 notes (standard reference, not scraped)

The one-sided flat function is C∞C^\inftyC∞ with identically zero Taylor series