This guide covers jump discontinuities and essential discontinuities. Key concepts include the definition and properties of each discontinuity type, including how to identify them using left-hand and right-hand limits. The guide provides examples, practice questions, a glossary of terms, and exam strategies. It emphasizes that these discontinuities are not removable and often relate to sudden changes or infinite limits.

#Study Notes on Discontinuities

#Jump Discontinuity

#Definition

A jump discontinuity occurs at a point where the value of a function makes a sudden 'leap' between two values.

#Properties

At a jump discontinuity:
- The limit from the left and the limit from the right both exist.
- However, they are not equal.
A jump discontinuity is not a removable discontinuity.

#Example

Key Concept

Consider the function $f$ defined by:

$f(x) = \begin{cases} x^3 - x^2 - 2x & \text{for } x < 1 \\ 8 - x^2 - 3x & \text{for } x \geq 1 \end{cases}$

The left-hand limit as $x$ approaches 1: ...

Consider the function $f(x) = \begin{cases} 2x + 3 & \text{if } x < 2 \\ -x + 8 & \text{if } x \geq 2 \end{cases}$ . Does $f(x)$ have a jump discontinuity at $x = 2$ ?

Yes, because the left-hand limit is 7 and right-hand limit is 6

No, the function is continuous at $x=2$

Yes, because the function is undefined at $x=2$

No, because the left-hand limit is 6 and right-hand limit is 7