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Continuity - Problem 1

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

To check if a function is continuous at a point, first check if the function exists at that point (if for that x, there exists an f(x)). Then, check if the limit as x approaches that point exists. Recall that this can be done by checking if the left- and right-hand limits are the same. Finally, for a function to be continuous, the value of the function at a point must be the same as the limit of the function at that point.

Let's take a look at an example. We have this function graphed here y equals f(x). I want to know why this function is not continuous at x equals -2 right here.

Well, remember that there are three conditions for continuity. The function has to be defined at the number in question. The limit as x approaches that number f(x) has to exist. Finally, that limit has to equal the value of the function at that number.

Now the number here is -2. Our problem is with the first condition f(a) doesn't exist. F(-2) does not exist. It is not defined. So the function is not continuous at -2.

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