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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 $ψ$ 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 $F (0) = 0$ , $F (x) = x + x^{2} ψ (1 / x)$ for $x \neq 0$ .

[L1]

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

[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 $f : A \to R$ 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 $R$ , with the dictionary to monotone sequences).

Verification

technique · contradiction

1.1

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

given

1.2

For $x \neq 0$ , $F^{'} (x) = 1 + 2 x ψ (1 / x) - ψ^{'} (1 / x)$ . Along reciprocal sequences at which $ψ^{'} > 2$ the derivative is eventually negative, while along reciprocal sequences at which $ψ^{'} < - 2$ 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

Nothing in the library uses this result yet.

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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 00 that is monotone on no neighbourhood of 00