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A function with positive derivative at $0$ that is monotone on no neighbourhood of $0$

Example

There is a differentiable function $F$ with $F'(0)>0$ that is not monotone on any neighbourhood of $0$ .

Facts & Assumptions

Given: A bounded periodic differentiable function $\psi$ whose derivative takes values above $2$ and below $-2$ , obtained by scaling A bounded $C^1$ periodic oscillator made from a quartic Hermite spline, and $F(0)=0$ , $F(x)=x+x^2\psi(1/x)$ for $x\ne0$ .

[L1]

Derivative algebra and the chain rule are 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 $(f \circ g)'(c) = f'(g(c))\,g'(c)$ .

[L2]

If a differentiable function is nondecreasing, every nonzero difference quotient has the corresponding weak sign and its derivative is nonnegative; for a nonincreasing function the derivative is nonpositive (The derivative $f'(c) = \lim_{x \to c} \frac{f(x) - f(c)}{x - c}$ of $f : A \to \mathbb{R}$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, Nondecreasing, increasing (strictly increasing), nonincreasing, decreasing, monotone and strictly monotone real functions on a subset of $\mathbb{R}$ , with the dictionary to monotone sequences).

Verification

technique · contradiction

1.1

Boundedness of $\psi$ gives $F'(0)=\lim_{x\to0}(1+x\psi(1/x))=1>0$ .

given

1.2

For $x\ne0$ , $F'(x)=1+2x\psi(1/x)-\psi'(1/x)$ . Along reciprocal sequences at which $\psi'>2$ the derivative is eventually negative, while along reciprocal sequences at which $\psi'<-2$ it is eventually positive.

L1given

2.1

If $F$ were monotone on some neighbourhood, [L2] would force one weak derivative sign throughout it, contradicting step 1.2.

assume-contrastep 1.2L2discharge-contradiction∎

Depends on

Used by

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Dependency tree · next 3 levels

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Sources

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)

A function with positive derivative at 000 that is monotone on no neighbourhood of 000