Lecture 13: Points of Discontinuity and Their Classification

13.1 Discontinuity and Its Classification

13.1.1 Definition

#####Definition: Discontinuity

Description: If $f(x)$ is defined in a punctured neighborhood of $x_0$ but is not continuous at $x_0$, then $x_0$ is a discontinuity of $f$.

Explanation

• 1. The function must be defined in some punctured neighborhood of $x_0$.
• 2. If it is not continuous at $x_0$, then $x_0$ is a discontinuity.

Classification of Discontinuities

• 1. A discontinuity of the first kind: both one-sided limits exist and are finite.
• Removable discontinuity: the left- and right-hand limits are equal.
• Jump discontinuity: the left- and right-hand limits are unequal.
• 2. A discontinuity of the second kind: at least one one-sided limit does not exist as a finite number.
• Infinite discontinuity: for example, $f(x)=1/x$ at $x=0$.
• Oscillatory discontinuity: for example, $f(x)=\sin(1/x)$ at $x=0$.
• Infinite and oscillatory discontinuities are common cases, but they are not the only possibilities.

Situations That Commonly Require One-Sided Limits

• 1. Boundary points of piecewise-defined functions.
• 2. Expressions involving $e^x$ as $x\to\pm\infty$.
• 3. Expressions involving $\arctan x$ as $x\to\pm\infty$.

13.1.2 Example Problems

Example Question: For $f(x)=\dfrac{\ln|x|}{|x-1|}\sin x$, find and classify its discontinuities.

• Analysis
• The function is undefined at $x=0$ and $x=1$.
• At $x=0$, since $\sin x\sim x$ and $|x-1|\to1$,

Therefore, $x=0$ is a removable discontinuity.

• At $x=1$, since $\ln x\sim x-1$,

The one-sided limits are finite but unequal, so $x=1$ is a jump discontinuity.

• Question Type: Discontinuity