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 do one last example. Why is this function; y equals h(x), I have graphed here not continuous at x equals 4? Recall the definition of continuity.
There are three conditions. The function's got to be defined at the number in question in this case 4. The limit as x approaches that number of f(x) has to exist. The limit as x approaches a of f(x) has to equal the value at a.
Here, the function is actually defined at 4. This point indicates that f(4) is 5. The limit exists as well. As you approach 4 from either the right, or the left, the limiting value is 1. The problem is that these two things are not the same. The value of the function is 5. The limit of the function is 1.
So the problem is, that the limit as x approaches 4 of h(x) is 1. That's not equal to h(4) which is 5. So this violates condition 3 of the definition of continuity.