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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 44, is the
function
SOLUTION:
and
and
Figure
44.Function
Therefore,
the curve is continuous at x=2 EXAMPLE:
In figure 45, 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
45.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 46 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
46.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:
