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Name: Ryan Z.
Status: student	
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Date: 6/23/2004

How do you algebraically find out if a function is discontinuous?

Ryan Z.,

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
Physics Instructor
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.

There can 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.

Vince Calder

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