Continuity

Defintion of Continuity at a Point

A function is continuous at c if:

  1. f(c) is defined





    4.
Types of Discontinuity

A function has a removable discontinuity when the function can be made continuous by filling in ONE point on the graph.

Where are the removable and nonremovable discontinuities in the graph above?

Answer: x = 0 is a removable discontinuity. x = -1 and x = 1 are nonremovable discontinuities.

Investigate the graph of  

Answers: There is a point of discontinuity at x = 1.
               It is removable.
               A new function is  f(x) = x - 3

Look at the graph of   f(x) = |x|
                                             x
This is an example of nonremovable discontinuity. The point of discontinuity is at x = 0.

One-Sided Limits

Investigate 

The .  There is no  because the function does not exist anywhere on the

left of x = -3.
 
 

Investigate   .    The limit as x approaches 0 from the left is -1.
                

Investigate   .    The limit as x approaches 0 from the right is 0.
 

Definition of a Continuous Function

A function f is continuous on the closed interval [a,b] if it is continuous on the open interval (a,b) and

  

Investigate  

Is f(x) continuous over [-1,3]?

Answer:   yes

Properties of Continuity

If b is a real number and f and g are continuous at x = c, then the following functions are also continuous at c.
  1. bf
  2. f + g
  3. f - g
  4. fg
  6. f(g(x))
Intermediate Value Theorem

If f is continuous on [a,b] and k is any number between f(a) and f(b), then there is at least one number c in [a,b] such that f(c) = k.

Problems