thomas.ross
thomas.ross 5d ago โ€ข 0 views

Understanding Discontinuities with Formal Definition Examples

Hey there! ๐Ÿ‘‹๐Ÿฝ Ever tripped over a discontinuity in math? ๐Ÿค” Don't worry, it happens to the best of us! Let's break down what discontinuities are with some clear examples and a handy quiz to test your understanding. Get ready to level up your math skills!
๐Ÿงฎ Mathematics

1 Answers

โœ… Best Answer
User Avatar
jesus_serrano Dec 31, 2025

๐Ÿ“š Quick Study Guide

    ๐Ÿ” A discontinuity occurs when a function is not continuous at a particular point. This means there's a break, jump, or hole in the graph at that point.
    ๐Ÿ”ข Formal Definition: A function $f(x)$ is discontinuous at $x=c$ if any of these conditions are met:
  • ๐Ÿšซ $f(c)$ is not defined.
  • ๐Ÿ“ˆ $\lim_{x \to c} f(x)$ does not exist.
  • โ‰  $\lim_{x \to c} f(x) \neq f(c)$.
    ๐Ÿ’ก Types of Discontinuities:
  • โœ‚๏ธ Removable Discontinuity: A hole in the graph, where the limit exists but doesn't equal the function's value at that point.
  • ๐Ÿฆ˜ Jump Discontinuity: The function 'jumps' from one value to another, so the left and right limits exist but are not equal.
  • ๐Ÿ’ฅ Infinite Discontinuity: The function approaches infinity (or negative infinity) as $x$ approaches $c$. This often occurs at vertical asymptotes.

Practice Quiz

  1. What condition MUST be met for a function $f(x)$ to be continuous at $x=c$?
    1. A) $f(c)$ must be defined.
    2. B) $\lim_{x \to c} f(x)$ must exist.
    3. C) $\lim_{x \to c} f(x) = f(c)$.
    4. D) All of the above.
  2. Which type of discontinuity is characterized by a 'hole' in the graph?
    1. A) Jump Discontinuity
    2. B) Infinite Discontinuity
    3. C) Removable Discontinuity
    4. D) Essential Discontinuity
  3. For the function $f(x) = \frac{x^2 - 4}{x - 2}$, what type of discontinuity exists at $x = 2$?
    1. A) Jump Discontinuity
    2. B) Infinite Discontinuity
    3. C) Removable Discontinuity
    4. D) No Discontinuity
  4. What happens to the function near an infinite discontinuity?
    1. A) It approaches a constant value.
    2. B) It approaches zero.
    3. C) It approaches infinity (or negative infinity).
    4. D) It oscillates between two values.
  5. If $\lim_{x \to c^-} f(x) = 3$ and $\lim_{x \to c^+} f(x) = 5$, what type of discontinuity exists at $x = c$?
    1. A) Removable Discontinuity
    2. B) Jump Discontinuity
    3. C) Infinite Discontinuity
    4. D) No Discontinuity
  6. Which of the following functions has a jump discontinuity at $x=0$?
    1. A) $f(x) = x^2$
    2. B) $f(x) = \frac{1}{x}$
    3. C) $f(x) = \begin{cases} 1, & x < 0 \\ 2, & x \geq 0 \end{cases}$
    4. D) $f(x) = |x|$
  7. For a function to have a removable discontinuity at $x=c$, which of the following must be true?
    1. A) The limit exists, and $f(c)$ is defined, but they are not equal.
    2. B) The limit does not exist.
    3. C) $f(c)$ is not defined.
    4. D) The function approaches infinity.
Click to see Answers
  1. D
  2. C
  3. C
  4. C
  5. B
  6. C
  7. A

Join the discussion

Please log in to post your answer.

Log In

Earn 2 Points for answering. If your answer is selected as the best, you'll get +20 Points! ๐Ÿš€