      Custom Search   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:    Integrated Publishing, Inc. - A (SDVOSB) Service Disabled Veteran Owned Small Business