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!

Why It Is Still Essential Today

In the modern era, we have WolframAlpha and ChatGPT. Why suffer through thousands of hand-solved limits and derivatives? demidovich calculus

Solving checklist (use for each problem)

• Restate clearly; specify domain and parameters.
• Identify goal (compute limit, prove convergence, find extremum).
• Try simple cases or symmetry; look for substitutions.
• Consider inequalities and dominant terms.
• Check endpoints, special parameter values, and existence conditions.
• Write final answer with justification and estimate error if approximate.

2. The Asymptotic Difficulty

Most American textbooks group problems by difficulty (Easy, Medium, Hard). Demidovich does something far more cunning. The difficulty rises smoothly but inexorably within a single subsection.

• Problem 1.1: Find the domain of y = sqrt(x-2). (Trivial)
• Problem 1.10: Find the domain of y = arcsin(log(x^2 - 1)). (Moderate)
• Problem 1.30: A piecewise function with nested radicals and absolute values. (Difficult)

The jump from problem 10 to problem 50 is exponential. This forces a non-linear learning curve. If you stop when it gets hard, you fail. You must push through the pain threshold.