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.