Continuous Functions (College Board AP® Calculus AB)

• Proving that a given function is continuous on all points of its domain can become quite tedious

  • Luckily, there are standard results that you can use for many commonly-occurring types of function

Polynomial functions

• A polynomial function is a function made up of sums or differences of positive integer powers of x, along possibly with constant terms

  • These are polynomial functions:

    • 

    • 

  • These are not polynomial functions:

    • 

      • Because is not a positive integer power of

    • 

      • Because is not a positive integer power of

• Polynomial functions are continuous on all points in their domains

  • This includes the domain of all real numbers

  • Or any smaller domain defined for a particular function

Rational functions

• A rational function is a function defined as a fraction (or quotient), where the expressions in the numerator and denominator are both polynomials

  • E.g. is a rational function

• A rational function is continuous everywhere, except at points where its denominator is equal to zero

  • E.g. is not continuous at or

  • However it is continuous over all other real numbers

  • Or any smaller domain not containing or

• The graphs of rational functions have vertical asymptotes at points where their denominators are equal to zero

Exponential functions

• An exponential function is a function of the form or , where and are non-zero real number constants, is a positive real number constant, and is the exponential constant

  • I.e. a function where the variable appears as an exponent (or power)

• Exponential functions are continuous on all points in their domains

  • This includes the domain of all real numbers

  • Or any smaller domain defined for a particular function

• The graphs of exponential functions of the forms given above have horizontal asymptotes at

Logarithmic functions

• A logarithmic function is a function of the form or , where and are non-zero real number constants, and is a positive real number constant

  • I.e. a function where the variable appears inside a logarithm

  • Remember that

• Logarithmic functions of the above forms are continuous at all points where

  • E.g. is continuous for all ; is continuous for all

  • A smaller domain on the 'correct side of zero' may also be defined for a particular function

• The graphs of logarithmic functions of the forms given above have vertical asymptotes at

Trigonometric functions

• Sine or cosine functions of the form or , where and are non-zero real number constants, are continuous on all points in their domains

  • This includes the domain of all real numbers

  • Or any smaller domain defined for a particular function

• A tangent function of the form , where and are non-zero real number constants, is continuous at all points where is not an odd multiple of

  • I.e. at all points where

  • At points where , a tangent function of the above form has a vertical asymptote