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 $0$ .

The standard flat function is smooth and flat at zero

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources