#Study Notes on Discontinuities

#Table of Contents

1. Jump Discontinuity
   • Definition
   • Properties
   • Example
2. Essential Discontinuity
   • Definition
   • Properties
   • Example
3. Practice Questions
4. Glossary
5. Summary and Key Takeaways
6. Exam Strategy

#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

• The left-hand limit as $x$ approaches 1: $$\lim_{{x \to 1^{-}}} f(x) = -2$$
• The right-hand limit as $x$ approaches 1: $$\lim_{{x \to 1^{+}}} f(x) = 4$$
• Since these limits are not equal, $f$ has a jump discontinuity at $x = 1$.

#Essential Discontinuity

#Definition

A function has an essential (or infinite) discontinuity at a point where:

• The limit from the left or the limit from the right (or both) do not exist, or are infinite.

#Properties

• An essential discontinuity is often associated with a vertical asymptote.
• It is not a removable discontinuity.

#Example

• The left-hand limit as $x$ approaches 5: $$\lim_{{x \to 5^{-}}} f(x) = 2 \quad (\text{and } f(5) = 2 \text{ as well})$$
• The right-hand limit as $x$ approaches 5: $$\lim_{{x \to 5^{+}}} f(x) = -\infty$$
• Because the right-hand limit is infinite, $f$ has an essential discontinuity at $x = 5$.

#Practice Questions

#Jump Discontinuity

Practice Question

1. Consider the function $g$ defined by: $$g(x) = \begin{cases} 2x + 1 & \text{for } x < 3 \\ -x + 7 & \text{for } x \geq 3 \end{cases}$$ Determine if $g$ has a jump discontinuity at $x = 3$.

#Essential Discontinuity

Practice Question

2. Consider the function $h$ defined by: $$h(x) = \begin{cases} \frac{1}{x-2} & \text{for } x < 2 \\ x^2 - 4 & \text{for } x \geq 2 \end{cases}$$ Determine if $h$ has an essential discontinuity at $x = 2$.

#Glossary

• Jump Discontinuity: A type of discontinuity where the function makes a sudden 'leap' between two values.
• Essential Discontinuity: A type of discontinuity where the limits do not exist or are infinite, often associated with vertical asymptotes.
• Limit: The value that a function approaches as the input approaches some value.
• Removable Discontinuity: A discontinuity that can be "removed" by redefining the function at that point.

#Summary and Key Takeaways

#Summary

• Jump Discontinuity: Occurs when the left-hand and right-hand limits exist but are not equal.
• Essential Discontinuity: Occurs when the limits do not exist or are infinite, often leading to vertical asymptotes.

#Key Takeaways

• Jump discontinuities indicate a sudden change in function values.
• Essential discontinuities often involve infinite limits.
• Both types of discontinuities are not removable.

#Exam Strategy

1. Identify Limits: Always check the left-hand and right-hand limits to identify the type of discontinuity.
2. Graph Analysis: Visualize the function graph to understand discontinuities better.
3. Practice: Solve various problems involving discontinuities to become familiar with different scenarios.