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
Therefore, the curve is continuous at
EXAMPLE: In figure 4-5, is the function
continuous at f(2)?
f(2) is undefined at
and the function is therefore discontinuous at
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.
In the following definitions of the functions, find where the functions are discontinuous and then extend the definitions so that the functions are continuous: