The Science That Draws Necessary Conclusions

To Learn Math

Archive for October 2008

Continuity and Discontinuity

with one comment

A function is said to be continuous at a if \displaystyle \lim_{x \to a} f(x) = f(a). A continuous function’s points are all connected. Below are examples of functions that are not continuous.

The graph above is an example of jump discontinuity. There is jump continuity at each integer. The limit from the positive and negative direction of each is a finite number, but they are not equal. Let’s look at the point where x=2. \displaystyle \lim_{x \to 2^-}y=1 and \displaystyle \lim_{x \to 2^+}y=2. Because both numbers are not infinite and are not equal, there is jump discontinuity where x=2.

The graph above is an example of infinite discontinuity. This is when the limit of a point from the negative or positive side is infinite or does not exist. There is infinite discontinuity at the point where x=0. \displaystyle \lim_{x \to 0^-}y=\infty and \displaystyle \lim_{x \to 0^+}y=\infty. Even though the limit from both sides equal infinity, the two lines never actually meet.

Above is an example of removable discontinuity. Removable discontinuity is when the limit from both sides of a point are equal and finite, but the function is undefinded at this point. There is a hole in the graph where x=-1. \displaystyle \lim_{x \to -1^-}y=-2 and \displaystyle \lim_{x \to -1^+}y=-2. Even though the limits are equal where x=-1, there is no point there.

Written by todizzle91

October 2, 2008 at 9:49 pm

Posted in Calculus