TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-30

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The standard flat function is smooth and flat at zero

Statement

The standard flat function $β$ is smooth on $R$ , and $β^{(n)} (0) = 0$ for every $n \in N_{0}$ .

Facts & Assumptions

Given: The standard flat function $β$ .

[F1]

The standard flat function is $0$ on $(- \infty, 0]$ and is $\exp (- 1 / t)$ on $(0, \infty)$ (The standard flat function).

[L1]

For every $m \in N$ , one has $β (t) / t^{m} \to 0$ as $t \to 0^{+}$ (Exponential decay dominates every inverse power near zero).

[L2]

The derivative of $\exp$ is $\exp$ , and one-variable derivatives satisfy the chain rule and algebra rules (The exponential function is smooth and $(\exp)^{'} = \exp$ , 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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ ).

[A1]

For each $n \in N_{0}$ there is a polynomial $P_{n}$ such that $β^{(n)} (t) = P_{n} (1 / t) \exp (- 1 / t)$ for every $t > 0$ .

Proof

technique · direct

1.1L2given

Repeatedly applying [L2] on $(0, \infty)$ proves [A1] by a routine induction on $n$ .

2.1L1step 1.1A1

For each $n$ , both $β^{(n)} (t)$ and $β^{(n)} (t) / t$ are finite linear combinations of terms $β (t) / t^{m}$ , so both tend to $0$ as $t \to 0^{+}$ by [L1] and step 1.1.

3.1F1step 2.1

We prove recursively that $β$ is $C^{n}$ , that $β^{(n)}$ vanishes on $(- \infty, 0]$ , and that $β^{(n)} (0) = 0$ . The case $n = 0$ is [F1]. If the claim holds for $n$ , then the left derivative of $β^{(n)}$ at $0$ is $0$ because $β^{(n)}$ is zero on $(- \infty, 0]$ , and the right derivative is $\lim_{t \to 0^{+}} β^{(n)} (t) / t = 0$ by step 2.1. Thus $β^{(n + 1)} (0)$ exists and equals $0$ , and step 2.1 also gives continuity at $0$ .

4.1step 3.1∎

Hence $β$ is smooth on $R$ and all of its derivatives vanish at $0$ .

Depends on

Used by

Positive-degree Dolbeault vanishing on pseudoconvex domains Corollary
A smooth nonanalytic solution of a first-order PDE Counterexample
Point evaluation is unbounded below the Sobolev continuity threshold Counterexample
The standard smooth step function Definition
The standard compactly supported bump on the line Example
The piecewise exponential flat function is not analytic at zero False statement
Existence of a smooth inhomogeneous dyadic frequency partition Lemma
For 0<p<1 the Hᵖ functional is a quasi-norm, and Hᵖ is a quasi-Banach space Remark

Dependency tree · two levels

24 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

John M. Lee, Introduction to Smooth Manifolds (standard reference, not scraped)
Will J. Merry, Differential Geometry (standard reference, not scraped)
Nigel Hitchin, Differentiable Manifolds (standard reference, not scraped)