DISCONTINUITIES

The discussion of discontinuties will be based on a comparison to continuity.

A function, f(x), is continuous at x = a if the following three conditions are met:

If a function f(x) is not continuous at x=a

then it is said to be discontinuous at x=a

We will use examples to show the above statements. EXAMPLE: In figure 4-4, is the function

SOLUTION:

and

and

Figure 4-4.-Function

Therefore, the curve is continuous at

x=2

EXAMPLE: In figure 4-5, is the function

continuous at f(2)?

SOLUTION:

f(2) is undefined at

x=2

and the function is therefore discontinuous at

x=2

Figure 4-5.-Function

However, by extending the original equation of f(x) to read

we will have a continuous function at

The value of 4 at

x = 2

was found by factoring the numerator of f(x) and then simplifying.

A common kind of discontinuity occurs when we are dealing with the tangent function of an angle. Figure 4-6 is the graph of the tangent as the angle varies from 0 to 90; that is, from 0 to . The value of the tangent at is undefined.

Thus the function is said to be discontinuous at 2.

Figure 4-6.-Graph of tangent function.

PRACTICE PROBLEMS:

In the following definitions of the functions, find where the functions are discontinuous and then extend the definitions so that the functions are continuous:

ANSWERS: