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AP®MathsCollege BoardCalculus ABStudy GuidesUnit 1: Limits & ContinuityContinuityNon-removable Discontinuities

Non-removable Discontinuities (College Board AP® Calculus AB)

Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Jump discontinuity

What is a jump discontinuity?

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

An example of the graph of a function with a jump discontinuity at x=a
Graph of a function with a jump discontinuity at x=a

  • Where there is a jump discontinuity

    • The limit from the left and the limit from the right both exist

    • But they are not equal

  • A jump discontinuity is not a removable discontinuity

  • For example, consider the function f defined by f open parentheses x close parentheses equals open curly brackets table row cell x cubed minus x squared minus 2 x comma space space x less than 1 end cell row cell 8 minus x squared minus 3 x comma space space x greater or equal than 1 end cell end table close

    • limit as x rightwards arrow 1 to the power of minus of f open parentheses x close parentheses equals negative 2

    • limit as x rightwards arrow 1 to the power of plus of f open parentheses x close parentheses equals 4

    • So f has a jump discontinuity at x equals 1

Essential discontinuity

  • 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

  • A common example is where the graph of a function has a vertical asymptote

Two examples of graphs of functions with essential discontinuities
Graphs of functions with essential discontinuities

  • An essential discontinuity is not a removable discontinuity

  • For example, consider the function f defined by f open parentheses x close parentheses equals open curly brackets table row cell x minus 3 comma space space x less or equal than 5 end cell row cell 4 minus fraction numerator 1 over denominator x minus 5 end fraction comma space space x greater than 5 end cell end table close

    • limit as x rightwards arrow 5 to the power of minus of f open parentheses x close parentheses equals 2 (and f open parentheses 5 close parentheses equals 2 as well)

    • But limit as x rightwards arrow 5 to the power of plus of f open parentheses x close parentheses equals negative infinity

    • Because the right-hand limit is infinite, f has an essential discontinuity at x equals 5

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    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths

    Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

    Dan Finlay

    Author: Dan Finlay

    Expertise: Maths Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.