Dr. Xiaozhou Li

Dr. Xiaozhou Li

Mathematics. Strength. Long-Term Mindset. Think Hard. Train Harder. Live Long.

Problem Sets: Continuity (连续性)

Published:

Last Update: 2025-10-24

Remark:

  • Per the sequence of the course outline, this set does not yet cover problems related to exponential functions, inverse functions, and logarithms—these will be included in subsequent sets.
  • 根据课程大纲的内容顺序安排，此部分练习暂未涵盖与指数函数、反函数及对数函数相关的题目 —— 这类题目将在后续练习中纳入。

Continuity

  • Let \(f(x)= \begin{cases}3+x^{2}, & x \leq 0, \\ \dfrac{\sin 3 x}{x}, & x>0 \end{cases}\). Is \( f(x) \) continuous?

  • Find the constants \(a\) and \(b\) for which the function \(f(x)= \begin{cases}\dfrac{\cos\pi x}{x^{2}+ax+b}, & x \neq \dfrac{1}{2}, \\ 2, & x = \dfrac{1}{2} ,\end{cases} \) is continuous at \( x = \dfrac{1}{2}\).

  • What is the number of removable discontinuities of the function \( f(x) = \dfrac{x - x^{3}}{\sin \pi x} \)?

  • What is the number of infinite discontinuities of the function \( f(x) = \dfrac{x^{2} - x}{x^{2} - 1}\sqrt{1 + \frac{1}{x^{2}}} \)?

  • Prove that \(f\) is continuous at \(c\) if and only if \(\lim_{h\to 0} f(c+h) = f( c ) \).

  • Prove that \(f(x) = \sin x\) is continuous at every point.

  • Prove that all six trigonometric functions are continuous wherever they are defined.

  • Show that the equation \(x^{3} - 15x -\cos x = 0\) has at least three solutions in the interval \([-4,4]\).

  • Suppose \(f(x)\) is continous on \([0,1]\), and for any \(x\in [0,1]\), we have \(0 \leq f(x) \leq 1\). Prove that there exists a point \(c \in [0,1]\), such that \(f( c ) = c\).

  • Show that the \(f(x)=\begin{cases}x^{2} \sin (1 / x), & x<0, \\ \sqrt{x}, & x\geq 0,\end{cases}\) is continuous on \((-\infty,\infty)\).

  • Let \(f(x)= \begin{cases}x^{\alpha}\sin\frac{1}{x^{\beta}}, & x \neq 0, \\ 0, & x=0 \end{cases}\). Find \(\alpha\) and \(\beta\) such that \( f(x) \) is continuous at \(x=0\).

  • The function \( f(x) = \begin{cases}\dfrac{1 - \cos\sqrt{x}}{ax}, & x > 0, \\b, & x \leq 0 \end{cases} \) is continuous at ( x = 0 ). Find \(a\) and \(b\).

  • Let the function \( f(x) = \begin{cases} -1, & x < 0, \\ 1, & x \geq 0, \end{cases} \) and \( g(x) = \begin{cases} 2 - ax, & x \leq -1, \\ x, & -1 < x < 0, \\ x - b, & x \geq 0. \end{cases} \). If \( f(x) + g(x) \) is continuous on \( \mathbb{R} \), then find \(a\) and \(b\).

  • Let the function \( f(x) = \begin{cases} x^{2} + 1, &x\leq c, \\ \dfrac{2}{x}, &x> c \end{cases} \) be continuous on \( (-\infty, \infty) \). Find \( c \).

  • Let \( f(x) \) be continuous on \( (-\infty, \infty) \), and \( f(f(x)) = x \). Prove that there exists at least one point \( \xi \in (-\infty, \infty) \) such that \( f(\xi) = \xi \).

  • Let \( f(x) = \begin{cases} x^{\lambda} \cos\dfrac{1}{x}, & x \neq 0, \\ 0, & x = 0, \end{cases} \). If \(f’\) is continuous at \( x = 0 \), find \( \lambda \).