Continuity · Definition

Types of discontinuity

calculuscontinuity

Before you start you need to know

A function is continuous at \(a\) if \(\lim_{x\to a}f(x)=f(a)\). If that equality fails, there is a discontinuity at \(a\).

Definition

A function is discontinuous at \(a\) when some continuity condition fails. It can be removable, a jump, or infinite.

\[\lim_{x\to a^-}f(x)\neq\lim_{x\to a^+}f(x)\ \Rightarrow\ \text{jump discontinuity}\]

When any of the three continuity conditions fails, there is a discontinuity. They are classified by what fails: removable (the limit exists but differs from the value), jump (different one-sided limits) or infinite (the function blows up).

A removable one can be "fixed" by redefining the value at that single point; the others cannot.

Classifying them is an exercise in one-sided limits: compute both sides and compare with the function value.

Think about it

What kind of discontinuity does (x²−1)/(x−1) have at x = 1?

Removable: the limit exists (2) but the function is not defined at 1.

Exercises 10

  1. What type of discontinuity does \(f(x)=\dfrac{x^{2}-1}{x-1}\) have at \(x=1\)?
  2. What type of discontinuity does \(f(x)=\dfrac{|x|}{x}\) have at \(x=0\)?
  3. What type of discontinuity does \(f(x)=\dfrac{1}{x-2}\) have at \(x=2\)?
  4. What type of discontinuity does \(f(x)=\dfrac{\sin x}{x}\) have at \(x=0\)?
  5. What type of discontinuity does \(f(x)=\dfrac{1}{x^{2}}\) have at \(x=0\)?
  6. A function has \(\lim_{x\to 2}f(x)=5\) but \(f(2)=3\). What type of discontinuity does it have at \(x=2\)?
  7. What type of discontinuity does \(f(x)=\dfrac{x^{2}-4}{x-2}\) have at \(x=2\)?
  8. Let \(f(x)=\begin{cases}x+1 & x<0\\ x^{2}+2 & x\ge 0\end{cases}\). What type of discontinuity does it have at \(x=0\)?
  9. What type of discontinuity does \(f(x)=\tan x\) have at \(x=\dfrac{\pi}{2}\)?
  10. What type of discontinuity does \(f(x)=\dfrac{\sin 3x}{x}\) have at \(x=0\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.