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Demidovich Calculus Instant

A classic!

Problem 2554 from Demidovich's "Problems in Mathematical Analysis"

Here's the problem:

Prove that the function

$$f(x) = \begincases x \sin \frac1x, & x \neq 0 \ 0, & x = 0 \endcases$$

is differentiable at $x=0$ and find $f'(0)$.

Solution

To prove differentiability at $x=0$, we must show that

$$\lim_h \to 0 \fracf(0+h) - f(0)h = \lim_h \to 0 \fracf(h)h$$

exists.

For $h \neq 0$,

$$f(h) = h \sin \frac1h$$

So,

$$\fracf(h)h = \sin \frac1h$$

Now,

$$\lim_h \to 0 \sin \frac1h$$

does not exist, but

$$\left| \sin \frac1h \right| \leq 1$$

for all $h \neq 0$. Hence,

$$\lim_h \to 0 \fracf(h)h = 0$$

exists and equals $0$. Therefore, $f'(0) = 0$.

The function $f(x)$ is differentiable at $x=0$, and $f'(0) = 0$.

However, $f(x)$ is not continuously differentiable at $x=0$ since $f'(x)$ does not exist for $x \neq 0$ or is not continuous at $x=0$ in a certain sense;

we could add more!

Boris Pavlovich Demidovich's " Problems in Mathematical Analysis

" is a legendary collection of over 3,000 exercises that has been a cornerstone of calculus and analysis education for decades. Known for its high difficulty compared to standard textbooks like Thomas Calculus, it is widely used by students and instructors for mastering rigorous mathematical techniques. Key Content Areas

The collection is organized into chapters that follow a traditional progression through higher mathematics:

Introduction to Analysis: Basics of functions, graphs, and the fundamental concept of limits.

Differentiation: Techniques for finding derivatives and their geometric applications, including extrema and curve sketching.

Integration: Extensive sections on indefinite and definite integrals, improper integrals, and applications such as calculating areas and volumes.

Series & Differential Equations: Advanced topics covering convergence of series and methods for solving ordinary differential equations.

Multivariable Calculus: Problems involving multiple and line integrals, change of variables, and functions of several variables. Why It Is Used

Rigorous Practice: It emphasizes proof-oriented reasoning and formal definitions rather than just computational exercises.

Comprehensive Resource: Each chapter typically starts with a brief theoretical introduction and worked-out typical problems to guide the student.

Global Recognition: Originally published in Russian, its translations have become a standard reference in technical schools and universities. demidovich calculus

You can find digital versions and community-led solutions on platforms like GitHub and Internet Archive. Problems In Mathematical Analysis : B. Demidovich (Ed.)

Boris Demidovich's Problems in Mathematical Analysis (often simply called "Demidovich"

) is a legendary fixture in mathematical education, particularly across Eastern Europe, China, and India. It is not a textbook in the traditional sense; it is a massive, rigorous collection of thousands of problems that has served as the "ultimate drill sergeant" for generations of aspiring physicists, engineers, and mathematicians. Mathematics Stack Exchange Why It Is Iconic Massive Volume: The most common edition contains over 3,000 problems

(some versions cite up to 5,000), covering everything from limits and single-variable derivatives to multivariable calculus, series, and differential equations. The "Russian School" Pedagogy:

It embodies the Soviet approach to mathematics—heavy on technical mastery, brute-force calculation, and a "gradual increase in difficulty" that forces students to build deep intuition through practice. Comprehensive Coverage:

It starts with the absolute basics of set theory and real numbers but quickly escalates into complex integration techniques that are rarely seen in modern Western introductory courses. Mathematics Stack Exchange The Solution Manual Mythos

One of the most famous aspects of "Demidovich" is the difficulty of finding English solutions. While the problem book was widely translated, the official detailed solution manuals are primarily available in Mathematics Stack Exchange The Chinese series, often titled 吉米多维奇数学分析习题集题解

, consists of six high-quality volumes that are highly sought after by students worldwide.

English-speaking students often rely on community-driven forums or older, fragmented manuals to check their work. Mathematics Stack Exchange Who Should Use It? Olympiad/Competitive Students:

For those preparing for exams like the Putnam or JEE, the problem sets offer a level of rigor that builds immense "mathematical stamina". Self-Learners:

If you find modern textbooks too "hand-holding," Demidovich provides a raw, unfiltered path to mastering the mechanics of calculus. Engineers/Physicists: The book focuses heavily on the

(the act of calculating) rather than just the abstract theory, making it a practical toolkit for those who need to solve real integrals and derivatives. Carnegie Mellon University Further Exploration Internet Archive

for scanned versions of the original Russian and translated English editions. Browse community discussions on Stack Exchange

regarding the best ways to navigate the numbering differences between various editions.

Explore a detailed breakdown of problem-solving strategies on from scholars who trained using these methods. versus modern calculus textbooks like How to Succeed in Calculus FAST FACTS

4. The Missing Answers (The Horror)

The original Soviet editions had no answers at the back. None. The translated versions often have "Answers and Hints" only for the odd-numbered problems, and even those are cryptic ("Yes," "No," "Converges conditionally"). This forces intellectual honesty. You cannot cheat. If you think you know the answer, you must prove it to a professor or a study group. This is the single most terrifying—and effective—pedagogical feature of the book.

How to Survive Demidovich (A Practical Guide)

If you are a student about to face this beast, do not go in blind. Here is the veteran’s protocol: A classic

  1. Do not buy the full edition alone. Get the "Solutions Manual" or the "Anton" textbook to read the theory. Demidovich is the gym; you need a coach (lecture notes) and a spotter (solution guide, used sparingly).
  2. The 20-Minute Rule. If you haven't made progress on a problem in 20 minutes, put a star next to it and move on. Return tomorrow. The subconscious mind solves what the conscious mind cannot.
  3. Collaborate. Soviet students worked in kruzhki (math circles). Get a study group. Do not cheat; discuss.
  4. Focus on the "Filter" Sections. If you can finish Limits (Part 2) and Indefinite Integrals (Part 5), you have conquered 80% of the difficulty. Differential equations and series become easy after that.
  5. Embrace the pain. When you feel stupid, remember that Grigori Perelman (who solved the Poincaré conjecture) likely struggled with problem 467 as a freshman.

The Legend of Demidovich: Why This Soviet-Era Problem Book Still Rules Calculus

If you walk into the dorm room of a math or physics student in Eastern Europe, China, or India, you will likely see a familiar sight on a cluttered desk: a thick, unassuming book with a blue cover, its pages yellowed and worn, the spine cracked from repeated use.

It is the Demidovich.

Formally titled Problems in Mathematical Analysis by B.P. Demidovich, this book is not a textbook. It is a rite of passage. For over half a century, this collection of problems has been the ultimate crucible for students learning calculus. It is known for one thing above all else: volume.

But in an age of sleek educational apps and "Calculus for Dummies," why are serious students still returning to this Soviet-era relic? The answer lies in the specific kind of mastery that Demidovich demands.

Part II: A Tour of the Torment (Structure of the Book)

The canonical edition is divided into eight major sections, each a deep dive into a specific realm of analysis. The numbering is legendary: "Demidovich 2,758" or "Demidovich 1,967" are often used as shorthand for specific classes of integrals or series among students.

1. Introduction to Analysis (Problems 1–310) This section deceives the careless. It begins gently: find the domain of a function, compute basic limits. But by problem 100, the limits become infamously tricky—involving nested radicals, exponentials of trigonometric functions, and careful use of equivalence of infinitesimals. It teaches the first hard lesson: nothing is trivial.

2. Differentiation (311–810) Derivatives of explicit, implicit, and parametric functions. While the early problems are mechanical, the middle sections introduce logarithmic differentiation on monstrous functions. The "prove that the function satisfies the differential equation" problems begin here, foreshadowing deeper physics applications.

3. Applications of Derivatives (811–1300) This is where many students break down. Problems involving tangent lines, curvature, and the Cauchy mean value theorem. But the true terror is the graphing section: given a complex rational function with parameters, determine asymptotes, intervals of monotonicity, concavity, and inflection points. Demidovich provides functions designed to have unexpected cusps or discontinuities that require L'Hôpital's rule several times.

4. Indefinite Integrals (1301–2260) Nearly 1,000 integrals, ordered by escalating cruelty.

The famous Demidovich integral—such as $\int \fracdxx^4+1$ or $\int \sqrt\tan x , dx$—is not a trick; it is a systematic application of method. The book provides no hand-holding, only the answer at the back.

5. Definite Integrals (2261–3100) Properties, the Newton-Leibniz formula, and improper integrals. The improper integrals (with parameters) are legendary for their subtlety: proving convergence conditionally, or using comparison tests with cleverly chosen divergent bounds. The section on Wallis' formula and the Gamma function is a student's first encounter with non-elementary integrals.

6. Multiple and Line Integrals (3101–4000) A significant jump in abstraction. Double and triple integrals over complex regions (cardioids, lemniscates). Change of variables in polar, cylindrical, and spherical coordinates—but with Jacobians that do not simplify nicely. Line integrals (first and second kind) test understanding of path independence and Green’s theorem.

7. Series and Sequences (4001–4400) Numerical series: every convergence test (D’Alembert, Cauchy, Raabe, Kummer, Gauss) is required. Power series: radius of convergence, summation, and expansion of functions. Fourier series on arbitrary intervals, including expansions with odd/even extensions that produce discontinuities requiring the Gibbs phenomenon.

8. Differential Equations (4401–4628) A concise introduction to first-order ODEs (exact, Bernoulli, Clairaut) and linear second-order with constant coefficients. While not a full ODE course, it demands the ability to solve quickly—often using integrating factors discovered by inspection.

Part I: The Man and the Method

Boris Pavlovich Demidovich (1906–1977) was a Soviet mathematician specializing in ordinary differential equations and dynamical systems. He was a professor at the elite Lomonosov Moscow State University (MGU), specifically within the Faculty of Physics and Mechanics.

The Soviet school of mathematics was famous for a specific pedagogical philosophy: mastery through immense, deliberate practice. The idea was not just to understand a theorem but to develop an almost tactile intuition for its application. A student should be able to "smell" a convergent series or "feel" a discontinuity. To achieve this, a textbook was insufficient; one needed a tank of problems.

Demidovich compiled his collection in the 1960s, drawing on decades of oral examination tradition and problem sets from MGU seminars. The result was a systematic, almost exhaustive catalog of every conceivable obstacle in single and multi-variable calculus. How to Survive Demidovich (A Practical Guide) If

Criticism

The book is not without faults.