Determining Discontinuous Functions
Name: Ryan Z.
How do you algebraically find out if a function is discontinuous?
Although I am not aware of any one perfect technique, there are some methods
that can provide information about whether a function is discontinuous. For
a function composed entirely of products and quotients of polynomials, look
for undefined points, zeroes in the denominator. A simple example of this
is f(x)=1+1/(x-1). There is a zero in a denominator at x=1. This is where
the function is discontinuous. For functions involving anything more
complex, you must analyze the properties of the individual functions within
the whole function and how they combine. For example, we know that the
tangent function is discontinuous. Any function with the tangent function
in it has a good chance of being discontinuous. Look at the tangent
discontinuities to see whether the entire function somehow manages to
counter the effects.
Dr. Ken Mellendorf
Illinois Central College
This is not so simple a question as it first appears. A discontinuous
function is one that has a "break" in the curve, that is easy to "say" but
as your question implies how does one "prove" that assertion, and that is
not quite so straightforward. In the first place there are different "kinds"
of discontinuities. The "simplest" one is a curve that has a "break". For
example, Y= 2*X for X less than1 and Y=4*X for X = 1 or X>1. Here the function is
defined for all values of X. There is another kind of discontinuity where
this is not the case. For example, Y=1 for X < 0 and
Y= 0 for X = 0 or X > 0. Here the two pieces are not connected. Another type
of similar discontinuity is the same except leave unspecified what happens
to Y when X=0. Then there is a missing point in the discontinuity. Another
type of discontinuity is one where the value of Y approaches + infinity for
some value of X greater than a specified value and - infinity for X less
than a certain value. An example is 1/sin(X) for X >0 and X less than 0.
even be some pathological discontinuities such as Y= 1 if X is an even integer
and Y = -1 if X is an odd integer. and Y= 0 for all other values of X.
Lastly, there are some functions that "look like" they should be
discontinuous, but are not. An example is
Y(X)= [sin(X)]/X as X approaches 0. That limit is Y(X)=1! Surprise.
Usually, the subject of calculus develops a number of theorems for
determining continuity because that is an important concept in that branch
of mathematics, but calculus does not address all possible cases of
discontinuous mathematical functions.
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Update: June 2012