1.10 Exploring Types of Discontinuities

• What is a discontinuity in a function?

  A break, hole, or jump
• A removable discontinuity occurs when the limit from both sides exists and is equal.
• The function $f(x) =$$\frac{x^{2} - 4}{x - 2}$ has a removable discontinuity at x = 2
• What is a jump discontinuity in a function?

  Abrupt change in values
• In a jump discontinuity, the left and right limits exist but are not equal.
• The function f(x) = \begin{cases} x, & \text{if } x \leq 1 \\ x + 2, & \text{if } x > 1 \end{cases}</latex> has a jump discontinuity at x = 1
• What is an infinite discontinuity in a function?

  Function approaches infinity
• An infinite discontinuity is characterized by a vertical asymptote.
• The function f(x) = \frac{1}{x - 3}</latex> has an infinite discontinuity at x = 3
• Match the type of discontinuity with its description:

  Removable ↔️ Hole in the function

  Jump ↔️ Abrupt change in values

  Infinite ↔️ Approaches infinity
• When does a removable discontinuity occur in a function?

  Hole in the graph
• In a removable discontinuity, the limits from both sides must be equal.
• The function $f(x) =$$\frac{x^{2} - 9}{x - 3}$ has a removable discontinuity at x = 3
• A removable discontinuity occurs when a function has a hole at a specific point
• A removable discontinuity occurs when limits from both sides exist and are equal.
• What happens to a function at a removable discontinuity?

  It has a hole
• What are the two key conditions for a jump discontinuity?

  Limits exist but are unequal
• An infinite discontinuity occurs when a function approaches infinity
• What type of line is associated with an infinite discontinuity?

  Vertical asymptote
• A removable discontinuity can be made continuous by redefining the function value at the hole
• What is a defining characteristic of a jump discontinuity?

  Left and right limits differ
• An infinite discontinuity is associated with a vertical asymptote.
• Match the type of discontinuity with its description:

  Removable ↔️ Hole in the function

  Jump ↔️ Abrupt change in values

  Infinite ↔️ Function approaches infinity
• What is a discontinuity in a function?

  A point where it's not continuous
• A removable discontinuity occurs when a function has a hole at a point but can be made continuous by redefining the function value
• In a removable discontinuity, the limits from both sides exist and are equal.
• Give an example of a function with a removable discontinuity.

  $f(x) =$$\frac{x^{2} - 9}{x - 3}$
• A jump discontinuity occurs when the function abruptly jumps from one value
• In a jump discontinuity, the left and right limits exist but are not equal.
• Give an example of a function with a jump discontinuity.

  f(x) = \begin{cases} x, & \text{if } x \leq 2 \\ x + 1, & \text{if } x > 2 \end{cases}</latex>
• An infinite discontinuity occurs when the function approaches infinity
• An infinite discontinuity is characterized by a vertical asymptote.
• Give an example of a function with an infinite discontinuity.

  f(x) = \frac{1}{x - 4}</latex>
• A removable discontinuity occurs when the limit from both sides exists and is equal
• A removable discontinuity is visually characterized by a hole in the graph.
• What adjustment can be made to remove a removable discontinuity?

  Redefine the function value
• A jump discontinuity occurs when the limits from both sides exist but are not equal
• In a jump discontinuity, \lim_{x \to c^{ - }} f(x)</latex> and $\lim_{x \to c^{ + }} f(x)$ both exist.
• What is the key difference between jump and removable discontinuities?

  Left and right limits
• An infinite discontinuity occurs when the function approaches infinity