ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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The indicator of ${1, 1 / 2, 1 / 4, 1 / 8, \dots}$ is discontinuous at $0$ , but its integral function has derivative $0 = f (0)$ there

Example

Define $f : [0, 1] \to R$ by

$f (x) = {\begin{cases} 1, & x = 2^{- n} for some integer n \geq 0, \\ 0, & otherwise. \end{cases}$

Then $f$ is Riemann integrable with integral zero on every subinterval. Its integral function $F (x) = \int_{0}^{x} f$ is therefore identically zero, so $F^{'} (0) = 0 = f (0)$ , although $f$ is discontinuous at $0$ .

Facts & Assumptions

Given: The sparse-spike function $f$ .

[L1]

The geometric sequence $2^{- n}$ tends to $0$ (For $∣ r ∣ < 1$ the sequence $r^{k}$ is null, and for $∣ r ∣ > 1$ the sequence $∣ r ∣^{k}$ diverges to $+ \infty$ ).

[L2]

A bounded function is integrable exactly when, for every $ε > 0$ , some partition has upper-minus-lower sum below $ε$ (Riemann's criterion: a bounded $f$ on $[a, b]$ is Darboux integrable if and only if for every real $ε > 0$ there is a partition $P$ with $U (f, P) - L (f, P) < ε$ ).

[L3]

The integral function is $F (x) = \int_{0}^{x} f$ (The integral function $F (x) : = \int_{a}^{x} f$ of an integrable $f$ ).

Verification

technique · direct

1.1

The function is bounded between $0$ and $1$ , and every nondegenerate interval contains a point outside the countable spike set, so every lower Darboux sum is $0$ .

givenconstruct

1.2

Given $ε > 0$ , choose $N$ with $2^{- N} < ε / 2$ by [L1]. Put the finitely many spikes $1, 2^{- 1}, \dots, 2^{- (N - 1)}$ in partition intervals of total length below $ε / 2$ , and put all remaining spikes in $[0, 2^{- N}]$ . The resulting upper sum is below $ε$ .

givenL1construct

2.1

By [L2], $f$ is integrable, and steps 1.1--1.2 force its integral to be $0$ . The same construction after restriction gives integral $0$ on every subinterval.

step 1.1step 1.2L2

3.1

By [L3] and step 2.1, $F (x) = 0$ for every $x$ , so its relative derivative at $0$ is $0$ .

step 2.1L3

4.1

Along the spike sequence $2^{- n} \to 0$ , the values are $1$ , while $f (0) = 0$ ; thus $f$ is discontinuous at $0$ and $F^{'} (0) = 0 = f (0)$ .

givenstep 3.1L1∎

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Sources

Riemann integral (standard reference, not scraped)

The indicator of {1,1/2,1/4,1/8,…}{1,1/2,1/4,1/8,…} is discontinuous at 00, but its integral function has derivative 0=f(0)0=f(0) there