Continuity

Continuity

If the limit exists at x  =  c, the function has some nice properties.  However, even if the limit exists, there may still be a hole.  We define a function to be continuous at x  =  c is the limit exists and the function agrees with the limit at x  =  c.  More formally

 Definition of a Continuous Function A function is continuous at x = c if the the limit exists there and

A function is called continuous if it is continuous for all real numbers.  In other words, if the graph has no holes asymptotes, or ,breaks then the function is continuous.  If you can draw the function without lifting your pencil then it is continuous.  Below are some examples of continuous functions.

Continuous Functions

1)  Polynomials

2)  sin and cos

3)  Rational Functions where the denominator is nonzero

4)  Sums, Differences, and Products of continuous functions

5)  Quotients of continuous functions

6)  Compositions of continuous functions

Examples:

The following are continuous:

A)  y = x2 + 3x - 4

B)  y = x sin x

C)                1
y  =
1+ x2

Exercises:  Determine whether the following are continuous.  If they are not continuous, at which points are they discontinuous?  Hold the mouse on the yellow rectangle for the answer.

A)                  x - 1
y =
x + 1

B)                       x
y =
x2 + 3x - 4

C)
 y  =  { 2x + 3    for    x < 1 3x - 2    for    x > 1

D)
 y  =  { x2          for    x < 2 5x - 6    for    x > 2

E)  For what value of k is the function continuous?

 y  =  { 3x2 - 5    for    x < 1 5x - k     for    x > 1

Applications:

Suppose that your account initially has \$10,00 in it.  The account pays 5% annual interest compounded monthly.  Sketch the graph of your balance as a function of time.  Is it continuous?

Sketch the graph of the population of the earth as a function of time.  Is this a continuous function.  Why is it reasonable to represent this graph as a continuous function?

Sketch the graph of your telephone costs for using the phone.  Is this a continuous function?

Sketch the graph of your blood pressure as a function of time while on a bicycle ride.  Is this a continuous function?

Come up with a continuous function  and a discontinuous function that occurs in the real world.