Discontinuity of a function

Discontinuity of a Function

Definition:

  • Discontinuity: A point c in the domain of a function f(x) where the function fails to be continuous.

Types of Discontinuities:

  1. Removable Discontinuity:

    • A discontinuity at c that can be "filled in" or removed by redefining the function at that point.
    • Example: f(x)=x21x1, where the function is undefined but can be made continuous by defining f(1)=2.
  2. Jump Discontinuity:

    • A discontinuity where the function "jumps" from one value to another at a specific point.
    • Example: f(x)=sign(x) (signum function) at x=0, where the function transitions from -1 to 1 abruptly.
  3. Infinite Discontinuity:

    • A discontinuity occurring when the function's value becomes unbounded (tends to infinity) at a certain point.
    • Example: f(x)=1x at x=0, where the function's values tend to infinity as x approaches 0.
  4. Essential Discontinuity:

    • A discontinuity that doesn't fall into the categories of removable, jump, or infinite discontinuities.
    • Example: f(x)=sin(1x), displaying erratic oscillatory behavior around 0.

Graphical Representation:

  • Removable Discontinuity: The graph might have a hole or gap that can be filled by redefining the function at that point.

  • Jump Discontinuity: The graph displays a sudden jump from one point to another.

  • Infinite Discontinuity: The graph may have a vertical asymptote or an unbounded increase/decrease around a point.

Identifying Discontinuities:

  • Algebraic Analysis: Observing the function's behavior algebraically, like undefined values or changes in function values.

  • Graphical Examination: Looking for breaks, jumps, or erratic behavior in the graph of the function.