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

Example

Define $ϕ (x) = {\begin{cases} \exp (- 1 / x), & x > 0, \\ 0, & x \leq 0. \end{cases}$ Then $ϕ \in C^{\infty} (R)$ and $ϕ^{(m)} (0) = 0$ for every $m \in N$ , although $ϕ (x) > 0$ 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 $a_{n} = f^{(n)} (c) / ι (n!)$ 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: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ 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 $ϕ^{(m)} (x) = P_{m} (1 / x) \exp (- 1 / x)$ 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 $x \to 0^{+}$ .

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 smooth function not equal to its Maclaurin series Counterexample
A nonzero smooth compactly supported bump Example

Dependency tree · two levels

40 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

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∞ with identically zero Taylor series