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A function differentiable on $[0,1]$ whose derivative is unbounded, hence not Riemann integrable

Statement refuted

False claim: if $G : [0,1] \to \mathbb{R}$ is differentiable at every point of $[0,1]$ (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), then $G'$ is Riemann integrable on $[0,1]$ (so that $\int_0^1 G' = G(1)-G(0)$ makes sense).

The claim is false. Put

$\phi(t) \;:=\; t^{2}(1-t)^{2} \qquad (t \in \mathbb{R}),$

a polynomial with $\phi(0)=\phi(1)=0$ and $\phi'(0)=\phi'(1)=0$ , and for $n \in \mathbb{N}$ set

$\alpha_n := \frac{1}{\iota(n+2)}, \qquad h_n := \frac{1}{\iota(n+2)^{4}}, \qquad c_n := \frac{1}{\iota(n+2)^{2}}, \qquad \beta_n := \alpha_n + h_n, \qquad I_n := [\alpha_n,\beta_n]$

(The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Integer powers $a^m$ ). The intervals $I_n$ are pairwise disjoint and lie in $(0,1]$ , and

$G : [0,1] \to \mathbb{R}, \qquad G(x) \;:=\; \begin{cases} c_n\,\phi\!\left(\dfrac{x-\alpha_n}{h_n}\right) & x \in I_n \text{ for some } n, \\[6pt] 0 & \text{otherwise} \end{cases}$

is differentiable at every point of $[0,1]$ , while

$G'\!\left(\alpha_n + \tfrac{1}{4}h_n\right) \;=\; \frac{3}{16}\,\iota(n+2)^{2} ,$

so $G'$ is unbounded on $[0,1]$ and therefore has no Darboux sums at all (For bounded $f$ on $[a,b]$ and a partition $P$ : the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$ ) and is not Riemann integrable.

The construction is entirely polynomial, and deliberately so. The classical witness is $x^{2}\sin(1/x^{2})$ ; the trigonometric functions are built on a later page of this library, so a bump glued from a single quartic is used instead. Only one bump is nonzero near any point of $(0,1]$ , so no series converges anywhere in the argument and no limit function is formed.

Facts & Assumptions

Given: The polynomial $\phi(t) = t^{2}(1-t)^{2}$ , the numbers $\alpha_n, h_n, c_n, \beta_n$ and intervals $I_n$ above, the function $G$ above, and a real $\varepsilon>0$ .

[L1]

Polynomial calculus: every polynomial function is differentiable at every real and continuous there, with $(t^{n})' = \iota(n)t^{\,n-1}$ for $n \ge 1$ , and sums, scalar multiples and products differentiate by the usual rules (For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, 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 $g(c) \ne 0$ , Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, 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, Integer powers $a^m$ ).

[L2]

Chain rule, and the derivative of an affine reparametrisation: $y \mapsto \psi((y-\alpha)/h)$ has derivative $\psi'((y-\alpha)/h)/h$ for $h \ne 0$ (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 $(f \circ g)'(c) = f'(g(c))\,g'(c)$ , 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 $g(c) \ne 0$ ).

[L3]

$\iota(n+2) \ge 2 > 1$ , $\iota$ is increasing on $\mathbb{N}$ , $\iota(m) < \iota(n)$ for $m<n$ , and for every real $w$ there is a natural $n$ with $w < \iota(n+2)$ ; also $\iota(n)^{k} \ge \iota(n)$ for $k \ge 1$ when $\iota(n)\ge 1$ (The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing, Every complete ordered field is Archimedean, For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ , Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$ , claims 3 and 4).

[L4]

If both one-sided limits of a function at $c$ exist and are equal to $L$ , the two-sided limit exists and equals $L$ (If $c$ is a limit point of the domain from both sides, the limit exists iff both one-sided limits exist and agree, The left and right limits of $f$ at $c$ , as limits of the restrictions of $f$ to $A \cap (-\infty, c)$ and $A \cap (c, \infty)$ , The $\varepsilon$ - $\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ ).

[L5]

Locality of limits: two functions on $A$ agreeing on a punctured neighbourhood of $c$ have the same limit behaviour at $c$ ; and a derivative survives restricting the domain provided the smaller domain still accumulates at the point (The limit at $c$ depends only on the restriction of $f$ to a punctured neighbourhood of $c$ , and passes to any subset of the domain having $c$ as a limit point, 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, Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ ).

[L6]

Every point of a nondegenerate interval is a limit point of it (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, Intervals of $\mathbb{R}$ : the nine order-convex forms, nondegeneracy, and length, Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ ).

[L7]

Darboux sums, and therefore Riemann integrability, require a bounded function (For bounded $f$ on $[a,b]$ and a partition $P$ : the infimum $m_i$ and supremum $M_i$ of $f$ on the $i$ -th subinterval, and the lower and upper Darboux sums $L(f,P) = \sum_i m_i \Delta_i$ and $U(f,P) = \sum_i M_i \Delta_i$ , Lower bound, bounded below, bounded set, The lower and upper Darboux integrals of a bounded $f$ on $[a,b]$ as $\sup_P L(f,P)$ and $\inf_P U(f,P)$ , Darboux integrability as their equality, and the notation $\int_a^b f$ ).

[L8]

A nonempty finite set of reals has a least element (Every nonempty finite set of reals has a maximum and a minimum, Maximum and minimum of a set).

[L9]

Ordered-field arithmetic: $t(1-t) \le 1/4$ for every real $t$ , since $(t - 1/2)^{2} \ge 0$ ; a positive real has a positive inverse; multiplying an inequality by a positive real preserves it; the order is total and transitive (Ordered field, Complete ordered field (least-upper-bound property), Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$ , claim 1).

[L10]

If $H$ is differentiable at every point of $[p,q]$ with $H'$ integrable there, then $\int_p^q H' = H(q)-H(p)$ (The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$ ).

Counterexample

technique · direct

1.1

By [L1], $\phi$ is differentiable everywhere with $\phi'(t) = 2t(1-t)^{2} - 2t^{2}(1-t) = 2t(1-t)(1-2t)$ ; in particular $\phi(0)=\phi(1)=0$ and $\phi'(0)=\phi'(1)=0$ , and $\phi'(1/4) = 2\cdot\tfrac14\cdot\tfrac34\cdot\tfrac12 = \tfrac{3}{16}$ .

L1L9

1.2

For $t \in [0,1]$ , $0 \le \phi(t) = \bigl(t(1-t)\bigr)^{2} \le (1/4)^{2} = 1/16$ , by [L9].

1.3

Each $\alpha_n, h_n, c_n$ is a positive real, $h_n \le \alpha_n$ and $c_n \le \alpha_n$ because $\iota(n+2) > 1$ , and $\alpha_n \le 1/2$ ; so $I_n \subseteq (0,1]$ , since $\beta_n = \alpha_n + h_n \le 2\alpha_n \le 1$ .

givenL3L9

2.1

The $I_n$ are pairwise disjoint. For $n \ge 1$ , $\alpha_{n-1}-\alpha_n = 1/\bigl(\iota(n+1)\iota(n+2)\bigr)$ while $h_n = 1/\iota(n+2)^{4}$ , and $\iota(n+2)^{4} > \iota(n+1)\iota(n+2)$ because $\iota(n+2)^{3} \ge 2\,\iota(n+2) > \iota(n+1)$ by [L3]; so $\beta_n < \alpha_{n-1}$ . Since $(\alpha_n)$ is strictly decreasing, $m < n$ gives $\beta_n < \alpha_{n-1} \le \alpha_m$ , and $I_n$ lies strictly below $I_m$ .

step 1.3L3L9

2.2

For each $n$ define the polynomial $\Phi_n(y) := c_n\,\phi\bigl((y-\alpha_n)/h_n\bigr)$ on $\mathbb{R}$ . By [L1] and [L2] it is differentiable everywhere with $\Phi_n'(y) = (c_n/h_n)\,\phi'\bigl((y-\alpha_n)/h_n\bigr)$ , and $\Phi_n(\alpha_n)=\Phi_n(\beta_n)=0$ , $\Phi_n'(\alpha_n)=\Phi_n'(\beta_n)=0$ by step 1.1.

step 1.1L1L2construct

2.3

Differentiability at a point of $(0,1]$ outside every $I_n$ . Let $x \in (0,1]$ with $x \notin I_m$ for every $m$ . By [L3] fix $N$ with $2/x < \iota(N+2)$ , so $2/\iota(N+2) < x$ ; for $m \ge N$ , $\beta_m \le 2\alpha_m = 2/\iota(m+2) \le 2/\iota(N+2) < x$ by step 1.3 and [L3].

step 1.3L3L9choose

3.1

So $G$ is a well-defined function on $[0,1]$ , no $x$ lying in two of the $I_n$ .

step 2.1given

3.2

Gaps around the endpoints. For each $n$ , $(\beta_{n+1}, \alpha_n)$ meets no $I_m$ : for $m \le n$ one has $\alpha_m \ge \alpha_n$ by step 2.1, and for $m \ge n+1$ one has $\beta_m \le \beta_{n+1}$ , again by step 2.1. Likewise $(\beta_n, \alpha_{n-1})$ meets no $I_m$ for $n \ge 1$ , and $(\beta_0, 1]$ meets none.

step 2.1L3

3.3

The finitely many closed intervals $I_0,\dots,I_{N-1}$ do not contain $x$ , so each of the positive reals $|x-\alpha_m|$ and $|x-\beta_m|$ for $m<N$ , together with $x - 2/\iota(N+2)$ , forms a nonempty finite set of positive reals; let $\rho$ be its least element, which is positive by [L8].

step 2.3L8L9choose

4.1

$G$ agrees with $\Phi_n$ on $I_n$ and with the zero function off $\bigcup_m I_m$ ; in particular $G(\alpha_n) = G(\beta_n) = 0$ and $G \ge 0$ everywhere.

step 3.1step 2.2L9

5.1

Differentiability at an interior point of a bump. Let $x \in (\alpha_n,\beta_n)$ . The difference quotients of $G$ and of $\Phi_n$ at $x$ agree on the punctured neighbourhood $0 < |y-x| < \min\{x-\alpha_n,\ \beta_n-x\}$ inside $[0,1]$ , so by [L5] and step 2.2, $G$ is differentiable at $x$ with $G'(x) = \Phi_n'(x)$ .

step 2.2step 4.1L5L6L8

5.2

Differentiability at a left endpoint $\alpha_n$ . On the right, $G$ agrees with $\Phi_n$ on $[\alpha_n,\beta_n]$ and $G(\alpha_n) = \Phi_n(\alpha_n)$ , so the right-hand limit of the difference quotient is $\Phi_n'(\alpha_n) = 0$ by [L5] and step 2.2. On the left, $G$ vanishes on $(\beta_{n+1},\alpha_n]$ by step 3.2 and step 4.1, so the quotient is identically $0$ there and its left-hand limit is $0$ . By [L4], $G'(\alpha_n) = 0$ .

step 2.2step 4.1step 3.2L4L5L6

5.3

Differentiability at a right endpoint $\beta_n$ . Symmetrically: on the left $G$ agrees with $\Phi_n$ , giving limit $\Phi_n'(\beta_n) = 0$ ; on the right $G$ vanishes on $[\beta_n,\alpha_{n-1})$ when $n \ge 1$ and on $[\beta_0,1]$ when $n = 0$ , by step 3.2, giving limit $0$ . By [L4], $G'(\beta_n) = 0$ .

step 2.2step 4.1step 3.2L4L5L6

5.4

Then $G$ vanishes on $(x-\rho,x+\rho)\cap[0,1]$ , so its difference quotient at $x$ is identically $0$ there and $G'(x) = 0$ by [L5] and [L6].

step 2.3step 3.3step 4.1L5L6

5.5

Differentiability at $0$ . For $y \in (0,1]$ : if $y \notin \bigcup_m I_m$ then $G(y)/y = 0$ ; and if $y \in I_m$ then $0 \le G(y) \le c_m/16$ by step 1.2 and step 4.1 and $y \ge \alpha_m$ , so $0 \le G(y)/y \le c_m/(16\,\alpha_m) = 1/\bigl(16\,\iota(m+2)\bigr)$ .

step 1.2step 4.1L9

6.1

Given $\varepsilon>0$ , fix by [L3] a natural $N$ with $1/\varepsilon < \iota(N+2)$ and put $\delta := \alpha_N > 0$ . If $0 < y < \delta$ and $y \in I_m$ then $\alpha_m \le y < \alpha_N$ , so $\iota(N+2) < \iota(m+2)$ by [L3], and step 5.5 gives $|G(y)/y| \le 1/\bigl(16\,\iota(m+2)\bigr) < \varepsilon$ ; otherwise $G(y)/y = 0$ .

step 5.5L3L9choose

6.2

$G'$ is unbounded. Put $u_n := \alpha_n + h_n/4$ , an interior point of $I_n$ ; by step 5.1 and step 2.2, $G'(u_n) = (c_n/h_n)\phi'(1/4) = \iota(n+2)^{2}\cdot\tfrac{3}{16}$ , using $c_n/h_n = \iota(n+2)^{4}/\iota(n+2)^{2} = \iota(n+2)^{2}$ . Given a real $M \ge 0$ , [L3] supplies $n$ with $\tfrac{16}{3}M < \iota(n+2) \le \iota(n+2)^{2}$ , so $G'(u_n) > M$ .

step 1.1step 2.2step 5.1L3L9

7.1

So the difference quotient of $G$ at $0$ , which is $y \mapsto G(y)/y$ on $(0,1]$ , tends to $0$ ; hence $G$ is differentiable at $0$ with $G'(0) = 0$ by [L6].

step 6.1L6

8.1

By steps 5.1, 5.2, 5.3, 5.4 and 7.1, $G$ is differentiable at every point of $[0,1]$ : every $x$ is either $0$ , or an interior point of some $I_n$ , or an endpoint of some $I_n$ , or a point of $(0,1]$ outside every $I_n$ .

step 5.1step 5.2step 5.3step 5.4step 7.1L9

9.1

Hence $G'$ is a function on $[0,1]$ that is not bounded, so it has no Darboux sums and is not Riemann integrable on $[0,1]$ by [L7]; the claim is false, and $\int_0^1 G'$ is an undefined symbol, so [L10] gives nothing here.

step 8.1step 6.2L7L10∎

Remarks

Only one bump is active near any point of $(0,1]$ , and that is what makes every step finite. The intervals $I_n$ accumulate only at $0$ , so a point of $(0,1]$ has a neighbourhood meeting at most one of them (steps 3.2 and 3.3), and the only place where infinitely many bumps are seen at once is the origin, where step 5.5 controls all of them by a single estimate. No series is summed anywhere.
The two exponents are what the construction turns on. Differentiability at $0$ needs $c_n/\alpha_n \to 0$ , and unboundedness of $G'$ needs $c_n/h_n \to \infty$ ; with $\alpha_n = 1/\iota(n+2)$ the choices $c_n = \alpha_n^{2}$ and $h_n = \alpha_n^{4}$ give $c_n/\alpha_n = \alpha_n \to 0$ and $c_n/h_n = \alpha_n^{-2} \to \infty$ . Any pair of exponents with the same two properties would do; these are verified explicitly in steps 6.1 and 6.2 because the construction is only as good as those two inequalities.
What this refutes and what it does not. It refutes the claim that every derivative is Riemann integrable, hence the naive reading of The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G' = f$ and $f$ is integrable, then $\int_a^b f = G(b)-G(a)$ with its integrability hypothesis deleted. It says nothing about whether $G'$ has a primitive — it does, namely $G$ — and nothing about the sharp class of functions for which $\int_a^b G' = G(b)-G(a)$ holds, which this library records but does not prove (Conventions of this page, and which sharpenings of the integral are taken up later in the reading order).

Depends on

Used by

FALSE: if u and v are differentiable on [a,b] then ∫ₐᵇ uv' = u(b)v(b)-u(a)v(a)-∫ₐᵇ u'v False statement

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Sources

Antiderivative (Wikipedia) (standard reference, not scraped)
Riemann integral (Wikipedia) (standard reference, not scraped)
J. M. H. Olmsted, Counterexamples in Analysis: Differentiation (standard reference, not scraped)

A function differentiable on [0,1][0,1][0,1] whose derivative is unbounded, hence not Riemann integrable