False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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FALSE: every smooth map between open subsets of the plane is real analytic

Statement

False claim: Every smooth map between open subsets of $R^{2}$ is real analytic.

Facts & Assumptions

Given: The function $ϕ : R \to R$ and planar map $F : R^{2} \to R^{2}$ defined by

$ϕ (x) : = {\begin{cases} \exp (- 1 / x^{2}), & x \neq 0, \\ 0, & x = 0, \end{cases} F (x, y) : = (ϕ (x), 0) .$

[L1]

The real exponential is $C^{\infty}$ and every derivative of it is the exponential itself (The exponential function is smooth and $(\exp)^{'} = \exp$ ).

[L2]

The derivative of a composite is given by the chain rule when the component derivatives exist (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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ ).

[L3]

Sums, scalar multiples, products, and quotients with nonzero denominator obey the corresponding derivative rules (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 $g (c) \neq 0$ ).

[L4]

For every natural $m$ and real $a > 0$ , $x^{m} / \exp (a x) \to 0$ as $x \to + \infty$ (The exponential dominates every fixed nonnegative integer power at $+ \infty$ ).

[L5]

A planar real function is $C^{k}$ when every coordinate-derivative word of length at most $k$ , including the word of length zero, exists and is continuous ( $C^{k}$ maps and multi-index derivative notation in Euclidean space).

[L6]

A smooth map $G = (u, v)$ is real analytic when, near every point, both components equal their total-degree Taylor series (Real-analytic maps between open subsets of the coordinate plane).

[L7]

The real exponential is positive everywhere and satisfies $\exp (- x) = 1 / \exp (x)$ (The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ).

Refutation

technique · direct

1.1

For every natural $m$ , repeated use of [L1], [L2], and [L3] gives a real polynomial $P_{m}$ such that $ϕ^{(m)} (x) = P_{m} (1 / x) \exp (- 1 / x^{2})$ for $x \neq 0$ : take $P_{0} = 1$ , and differentiation replaces $P_{m} (y)$ by the polynomial $- y^{2} P_{m}^{'} (y) + 2 y^{3} P_{m} (y)$ .

L1L2L3

2.1

As $x \to 0$ , every expression $P (1 / x) \exp (- 1 / x^{2})$ and its quotient by $x$ tends to $0$ : with $y = 1 / ∣ x ∣ \to + \infty$ , polynomial growth is bounded by a natural power of $y$ , which for $y \geq 1$ is bounded by a natural power of $y^{2}$ , and [L4] applied to $u = y^{2}$ makes that power times $\exp (- u)$ tend to zero.

step 1.1L4

3.1

Inductively set every derivative value $ϕ^{(m)} (0) = 0$ : step 2.1 makes $ϕ^{(m)}$ continuous at $0$ and makes its difference quotient there tend to $0$ , so the next derivative exists and has value $0$ . Thus $ϕ$ is smooth, and [L5] makes $F (x, y) = (ϕ (x), 0)$ smooth with every mixed derivative at $(0, 0)$ equal to $0$ .

step 2.1L5

4.1

By step 3.1, the total-degree Taylor series of $F$ at $(0, 0)$ is the zero map, but [L7] gives $F (x, 0) = (\exp (- 1 / x^{2}), 0) \neq (0, 0)$ for every $x \neq 0$ ; such points occur in every neighbourhood of the origin, so the equality required by [L6] fails there.

step 3.1L6L7

5.1

The map $F$ is smooth by step 3.1 and not real analytic by step 4.1, so it refutes the false claim.

step 3.1step 4.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

34 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

Michael Taylor, Introduction to Analysis in Several Variables, Ch. 2 §2.2, Exercise 4 (standard reference, not scraped)