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Continuity (College Board AP® Calculus AB): Study Guide

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Updated on

College Board Calculus ABCollege Board Calculus BC

Continuity at a point

What does continuous mean in the context of functions?

  • An informal way to think about what continuous means for functions is that 'a function is continuous anywhere you can draw its graph without lifting your pencil off the paper'

Graphs showing examples of functions that are continuous everywhere, and of functions that are not continuous at a single point

  • For the purposes of calculus, however, we need to work with a more formal definition!

What is the definition for a function to be continuous at a point?

  • A function f is continuous at the point x equals c if

    • f open parentheses c close parentheses exists,

    • limit as x rightwards arrow c of f open parentheses x close parentheses exists,

    • and limit as x rightwards arrow c of f open parentheses x close parentheses equals f open parentheses c close parentheses

  • That is

    • The function must have a well-defined finite value at x equals c

    • The function must have a well-defined finite limit at x equals c

      • This means that the one-sided limits must be equal at x equals c

    • The value of the limit and the value of the function must be equal at x equals c

  • Consider the functions f, g and h defined by

    • f open parentheses x close parentheses equals 1 over x

    • g open parentheses x close parentheses equals x over x

    • h open parentheses x close parentheses equals open curly brackets table row cell x comma space space x not equal to 0 end cell row cell 1 comma space space x equals 0 end cell end table close

  • For function f

    • 1 over 0 is not defined, so f open parentheses 0 close parentheses does not exist

    • f becomes unbounded as x approaches zero from the left and right, so limit as x rightwards arrow 0 of f open parentheses x close parentheses doesn't exist

    • For either of those reasons, f is not continuous at x equals 0

  • For function g

    • limit as x rightwards arrow 0 to the power of minus of g open parentheses x close parentheses equals limit as x rightwards arrow 0 to the power of plus of g open parentheses x close parentheses equals 1, so limit as x rightwards arrow 0 of g open parentheses x close parentheses equals 1

    • But 0 over 0 is not defined, so g open parentheses 0 close parentheses doesn't exist

    • Therefore g is not continuous at x equals 0

  • For function h

    • h open parentheses 0 close parentheses equals 1

      • so the function has a well-defined finite value at x equals 0

    • limit as x rightwards arrow 0 to the power of minus of h open parentheses x close parentheses equals limit as x rightwards arrow 0 to the power of plus of h open parentheses x close parentheses equals 0, so limit as x rightwards arrow 0 of h open parentheses x close parentheses equals 0

      • so the function has a well-defined finite limit at x equals 0

    • But limit as x rightwards arrow 0 of h open parentheses x close parentheses not equal to h open parentheses 0 close parentheses

    • Therefore h is not continuous at x equals 0

How does continuity work for a piecewise-defined function?

  • The definition of continuity at a point also applies to piecewise-defined functions

  • In practical terms this means that

    • At a boundary to a partition of the function's domain

      • A piecewise-defined function will be continuous if the following are all equal:

        • The value of the expression defining the function to the left of the boundary

        • The value of the expression defining the function to the right of the boundary

        • The value of the function at the boundary

  • This is not an alternative definition of continuity!

    • It is merely a 'side-effect' of the definition given above

    • See the worked example for an example of this

How does continuity work for combinations of functions?

  • You can use the following continuity theorem for combinations of functions:

    • If functions f and g are continuous at a point x equals a, then the following functions are also continuous at x equals a:

      • f plus g

      • f minus g

      • f times g

      • f over g (so long as g open parentheses a close parentheses not equal to 0)

Worked Example

(a) Explain why the function f defined by f open parentheses x close parentheses equals fraction numerator x plus 3 over denominator x minus 1 end fraction is not continuous at x equals 1.

Answer:

The main problem here is that f is not defined when x equals 1

f open parentheses 1 close parentheses equals fraction numerator 1 plus 3 over denominator 1 minus 1 end fraction equals 4 over 0 which is not defined

Alternatively, you could show that the limits from the left and right are unbounded at x equals 1, which also makes the function discontinuous at that point

f open parentheses 1 close parentheses does not exist, so f is not continuous at x equals 1

(b) Explain why the function g defined by g open parentheses x close parentheses equals open curly brackets table row cell x plus 4 comma space space x less than 0 end cell row cell 2 comma space space x equals 0 end cell row cell open parentheses x minus 2 close parentheses squared comma space space x greater than 0 end cell end table close is not continuous at x equals 0.

Answer:

First look at the left and right limits at x equals 0

limit as x rightwards arrow 0 to the power of minus of g open parentheses x close parentheses equals open parentheses 0 close parentheses plus 4 equals 4

limit as x rightwards arrow 0 to the power of plus of g open parentheses x close parentheses equals open parentheses open parentheses 0 close parentheses minus 2 close parentheses squared equals 4

Those are equal, so the limit exists

limit as x rightwards arrow 0 of g open parentheses x close parentheses equals 4

Now look at the value of the function at x equals 0

g open parentheses 0 close parentheses equals 2

So both the value of the function and the value of its limit are well-defined at x equals 0

The problem is that those two values are not equal

limit as x rightwards arrow 0 of g open parentheses x close parentheses not equal to g open parentheses 0 close parentheses, therefore g is not continuous at x equals 0

Continuity over an interval

What does it mean for a function to be continuous over an interval?

  • A function is continuous over an interval if it is continuous at every point in the interval

    • For example, we have seen that g open parentheses x close parentheses equals x over x is not continuous at x equals 0

    • But it is continuous over any interval that doesn't include 0

What does it mean for a function to be continuous?

  • A function is said to be continuous if it is continuous at every point in its domain

    • For example, we have seen that f open parentheses x close parentheses equals 1 over x is not continuous at x equals 0

      • Therefore the function f is not continuous over all the real numbers

    • However if we define the function space j as space j open parentheses x close parentheses equals 1 over x comma space space x not equal to 0

      • We have removed the discontinuity from its domain

      • Therefore space j is a continuous function

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    Previous:Selecting Procedures for Determining LimitsNext:Removable Discontinuities
    Author
    Reviewer
    Roger B

    Author: Roger B

    Expertise: Maths Content Creator

    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

    Reviewer: Dan Finlay

    Expertise: Maths Subject 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.