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Continuity - Problem 2 5,566 views

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 another example. I want to ask the question why is this function not continuous at x equals 1? Now recall the conditions for continuity.

They are three of them. First, the function f has to be defined at the point a, in this case 1. Second, the limit as x approaches a of f(x) has to exist. Third, the limit as x approaches a of f(x) has to equal f(a) , the value of the function at a.

Now here the number in question is 1. G(1) does exist, so the first condition is met. Let's take a look at the second condition. Does the limit as x approaches 1 exist?

Well, as x approaches 1 from the left, the value stays constant at 3. So that limit's 3. What about from the right? The limit as x approaches 1 from the right. We're coming along this branch here. We're coming down towards -2. So that limit is -2.

These limits aren't equal. So that means, the limit as x approaches 1 of g(x) does not exist. That means that this function violates condition 2 for continuity. So it's not continuous, because the limit as x approaches 1 of g(x) doesn't exist.