Objectives: In this tutorial, the definition of a function is continuous at some point is given. It is noted that this definition requires the checking of three conditions. Some examples applying this definition are given. Several theorems about continuous functions are given. Some examples of functions which are not continuous at some point are given the corresponding discontinuities are defined.

After working through these materials, the student should be able

• to determine symbolically whether a function is continuous at a given point;
• to apply the limit theorems to obtain theorems about continuous functions;
• to apply the theorems about continuous functions;
• to determine whether a piecewise defined function is continuous;
• to become aware of problems of determining whether a given function is conti nuous by using graphical techniques.

Modules:

Definition. A function f is continuous at x = a if and only if

• If a function f is continuous at x = a then we must have the following three conditions.
  1. f(a) is defined; in other words, a is in the domain of f.
  2. The limit
     

     must exist.

  3. The two numbers in 1. and 2., f(a) and L, must be equal.

• Examples.
  • at x = 3

    Discussion.

  • at x = 0

    Discussion.

  • at x = 3

    Discussion.

  • at x = 3

    Discussion.

  Definition. Suppose that we have a function like either f or h above which has a discontinuity at x = a such that it is possible to redefine the function at this point as with k above so that the new function is continuous at x = a. Then we say that the function has a removable discontinuity at x = a.

• From limit theorems B.13 and B.14, we have the following theorem.

  Theorem A. A polynomial is continuous at each real number. A rational function is continuous at each point of its domain.

• Also, from the other limit properties, we have the following theorem.

  Theorem B. Suppose that f and g are functions which are continuous at the point x = a and suppose that k is a constant. Then The product k f is continuous at x = a. The sum f + g is continuous at x = a. The difference f - g is continuous at x = a. The product f g is continuous at x = a. The quotient f / g is continuous at x = a provided that g(a) is not zero.

• Discussion

  Theorem C. Suppose that g is a function which is continuous at x = a and suppose that f is a function which is continuous at x = g(a) then the composition of f and g is continuous at x = a.

• Discussion [Using Flash]

• Examples.

  • at x = 2

    Discussion

  • at x = 2

    Discussion

• Quiz on checking whether functions are continuous. [Work only the basic quiz!]

  Definition. A function f is continuous on the interval I if it is continuous at each point of I. If I has an endpoint a then f is continuous at a if if a is a left-hand endpoint if a is a right-hand endpoint.

• Example:
  • 

    Discussion [Using Flash]

• A LiveMath notebook on determining whether a piecewise defined function is continuous.

  Definition. Suppose that we have a function as in the previous notebook such that the right-hand limit and the left-hand limit exist but L and M are different. Then we say that the function has a jump discontinuity at x = a.

• Drill problems in determining when a piecewise defined function is continuous.

• A LiveMath notebook illustrating the problem of using computers to determine graphically whether a function is continuous.

• Using a graphing calculator to investigate a function with a removable discontinuity.
  
  • TI-86 Graphing Calculator [Using Flash]
  • TI-85 Graphing Calculator

  Theorem D. The following functions are continuous at each point of its domain: f(x) = sin(x) f(x) = cos(x) f(x) = tan(x) f(x) = ax for any real number a > 0. f(x) = ex f(x) = ln(x) If f is a function that is continuous at each point of its domain and if f has an inverse, then the inverse f-1 is also continuous at each point of its domain.

• Quiz on checking whether functions are continuous.