1.13 Removing Discontinuities

• What is a discontinuity in the context of functions?

  A break or jump
• There are three main types of discontinuities: point, jump, and infinite
• Removable discontinuities can be identified graphically as holes in the graph.
• How are removable discontinuities identified algebraically?

  By factoring and canceling
• The limit at a point of removable discontinuity can be used to find the value needed to remove
• What is the purpose of evaluating limits at removable discontinuities?

  To find the missing value
• Functions can be redefined to remove removable discontinuities
• Removing a discontinuity ensures that the function becomes continuous at that point.
• What defines a removable discontinuity?

  A hole in the graph
• A jump discontinuity occurs when the function "jumps" from one value
• What is the characteristic feature of an infinite discontinuity?

  Approaches infinity
• Give an example of a removable discontinuity that can be redefined to $f(x) =$$x +$$2$ at $x =$$2$.


  $f(x) =$$\frac{x^{2} - 4}{x - 2}$
• The function f(x) = \begin{cases} x & \text{if } x < 1 \\ x + 2 & \text{if } x \geq 1 \end{cases}</latex> has a jump discontinuity at x
• At what point does the function $f(x) =$$\frac{1}{x - 3}$ have an infinite discontinuity?


  $x =$$3$
• A function is continuous at a point if the limit exists, the function is defined, and the limit equals the function's value.
• A discontinuity occurs when a function's graph has a break or gap
• What type of discontinuity can be filled by redefining the function?

  Removable discontinuity
• A jump discontinuity involves a sudden change in the function's value.
• An infinite discontinuity occurs when the function approaches infinity
• Give an example of a function with a removable discontinuity at x = 1.

  f(x) = \frac{x^{2} - 1}{x - 1}</latex>
• A function is continuous if it is defined, the limit exists, and the limit equals the function's value.
• A discontinuity occurs when any of the conditions for continuity are not met
• What is the graphical representation of a removable discontinuity?

  Hole
• Match the type of discontinuity with its description:

  Removable ↔️ Can be removed by redefining

  Jump ↔️ Has different left and right limits

  Infinite ↔️ Function approaches infinity
• Removable discontinuities on a graph appear as holes.
• A jump discontinuity appears on a graph as a step-like jump
• What type of discontinuity results in a vertical asymptote on a graph?

  Infinite discontinuity
• A removable discontinuity occurs when a function's graph has a gap but the limit exists at that point.
• A removable discontinuity is graphically identified by looking for a hole
• What condition must be checked after finding a hole in a graph to confirm a removable discontinuity?

  The limit exists
• The function $f(x) =$$\frac{x^{2} - 1}{x - 1}$ has a removable discontinuity at x = 1.
• What is the graphical representation of a removable discontinuity on a function's graph?

  A hole
• Removable discontinuities can be eliminated by redefining the function at the point of discontinuity.
• On a graph, a removable discontinuity appears as a hole
• How does a jump discontinuity appear on a graph?

  Abrupt change in the graph
• What is the graphical representation of a removable discontinuity?

  A hole in the graph
• A removable discontinuity occurs when a function's graph has a gap at a specific point, but the limit exists
• The function $f(x) =$$\frac{x^{2} - 1}{x - 1}$ has a removable discontinuity at x = 1</latex>
• A removable discontinuity occurs when a rational function has a factor common to both the numerator and denominator
• Steps to identify a removable discontinuity algebraically

  1️⃣ Factor both the numerator and denominator

  2️⃣ Cancel any common factors

  3️⃣ Observe any remaining expressions where the denominator would be zero