Pathological derivatives

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Problem 1.4.4 from de Souza P.N., Silva J. "Berkeley Problems in Mathematics" [deSi04]More problems in Section 1.4 "Differential Calculus"

(1)(2)

(1)

Give an example of a differentiable function $f: \mathbb {R} \rightarrow \mathbb {R}$ whose derivative $f'$ is not continuous.

(2)

Let $f$ be as in Part 1. If $f'(0) < 2 < f'(1)$ , prove that $f'(x) = 2$ for some $x \in [0,1]$ .

Problem III.3.9 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.3 "Continuous and Differentiable Functions"

Let $F \subset \mathbb {R}$ be a closed set with no interior points. Prove that there exists a strictly increasing function $f \in C^{1}(\mathbb {R})$ such that $\left\{ x \in \mathbb {R} \mid f^{\prime }(x)=0\right\} =F$ .

Problem III.3.11 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.3 "Continuous and Differentiable Functions"

(a)(b)

Can the derivative of a function that is everywhere differentiable and not constant on an interval vanish

(a)

at all irrational points of this interval?

(b)

on a dense subset (every nonempty subinterval contains a root of the derivative)?

Does the answer change if the derivative is additionally required to be bounded?

Problem III.3.30 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.3 "Continuous and Differentiable Functions"

(a)(b)

Let $\alpha >0, f \in C^{1}(\Delta ), N_{0}=\left\{ x \in \Delta \mid f^{\prime }(x)=0\right\}$ be the set of critical points of the function $f$ . Prove that

(a)

if $f^{\prime } \in \operatorname {Lip}(\alpha , c)$ for every point $c \in \Delta$ , then

N_{0}=\left\{ x \in \Delta \mid f \in \operatorname {Lip}(1+\alpha , x)\right\} ;

(b)

if the function $f$ is twice differentiable on $\Delta$ , then

N_{0}=\left\{ x \in \Delta \mid f \in \operatorname {Lip}(2, x)\right\} .

Remark.

A function $f$ , defined on an interval $\Delta$ containing a point $c$ , is said to belong to the class $\operatorname {Lip}(\alpha , c)$ ( $\alpha >0$ ) if there exist positive numbers $L$ and $\delta$ such that $\left|f(x)-f(c)\right| \leqslant L\left|x-c\right|^{\alpha }$ for every point $x$ in the intersection $\Delta \cap (c-\delta , c+\delta )$ .

A point at which the derivative of a smooth function (in the case of functions of several variables, the gradient of a smooth function) vanishes is called a critical point of that function, and the value of the function at a critical point is called its critical value.

Problem VIII.3.7 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section VIII.3 "Summable Functions"

Can an everywhere differentiable function on $\left[0,1\right]$ have a derivative that is not summable on $\left[0,1\right]$ ?

Problem 9.4.40 from Petrovic J.S. "Measure and Integral: Theory and Practice" [Pet25]More problems in Section 9.4 "Absolutely Continuous Functions"

Give an example of a function $f:[a, b] \rightarrow \mathbb {R}$ that is differentiable on $[a, b]$ but its derivative is not integrable over $[a, b]$ .

Problem 2.7.15 from Rudin W. "Real and Complex Analysis" [Rud87]More problems in Chapter 2.7 "Differentiation"

Construct a monotonic function $f$ on $\mathbb {R}^{1}$ so that $f^{\prime }(x)$ exists (finitely) for every $x \in \mathbb {R}^{1}$ , but $f^{\prime }$ is not a continuous function.

References

[deSi04] de Souza P.N., Silva J. Berkeley Problems in Mathematics. 3rd ed., Springer, 2004.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.

[Pet25] Petrovic J.S. Measure and Integral: Theory and Practice. CRC Press, 2025.

[Rud87] Rudin W. Real and Complex Analysis. 3rd ed., McGraw-Hill, 1987.