 Let's look at a couple of examples of continuity of a function at a point, and we're going to look at this from two different perspectives. We're going to look at it from a graphical perspective, but also utilizing those three tests for continuity at a point that we have in calculus. So the first one we're going to consider is the function 1 over x plus 1 at x equals negative 1. Now hopefully the first thing that strikes you is that this function is going to be undefined at x equals negative 1 because that would give us a zero in the denominator. If we look at this from a graphical point of view, we know there would be a vertical asymptote at x equals negative 1. It's going to cross the y-intercept at 1, so our graph is going to look like this. And then we'll have one down here, like this. So this would be an example of an infinite discontinuity at x equals negative 1. Now let's look at it from using, though, those three tests for continuity at a point. The first test states that the function value at the x value in question needs to exist. So our first point we're going to address does f of negative 1 exist? Well that's a no, and that's really where this test fails. We know it does not exist because we know there's an asymptote there. So this is where that test for continuity that we use in calculus really falls apart. So therefore this function is not continuous at x equals negative 1, and there is an infinite discontinuity present. Let's look at another example. Let's consider the function x squared minus 1 over x minus 1 at the x value of 2. So if we were to look at this from a graphical point of view, this is one of those that you hopefully learned in algebra 2 or pre-calculus that you would factor the numerator. It's a difference of 2 squares, of course. When you consider the denominator we have x minus 1's that cancel out. Remember that tells us there's going to be a hole in the graph at the x value of positive 1. So if we were to take a look at the graph of this, really what all remains is in the numerator, x plus 1. So this is really just a linear equation that we have remaining, but at x equals 1 there's going to be a hole in that line. Let's try to make ourselves a nice graph. So we know because of the x plus 1, I think y equals mx plus b, it's going to cross the y-intercept at 1. Our slope of course is 1, so up 1 over 1, up 1 over 1, but right here is where the hole is at x equals 1. Otherwise it just continues there, up 1 over 1, we could go down 1 and to the left 1. So this is what our graph would look like. And there's that hole at x equals 1. Now in this case though, we're concerned with x equals 2. So let's go ahead and apply the three tests for continuity and see how those play out. So remember the first test is that the function value needs to exist. So we need to know that f of 2 exists. Well we can see from our graph that it does, if you substitute 2 into the equation, into the function, we get 3. And you can see that function value right here. So the next step in the test is that the limit as you approach this x value of 2 needs to exist. Now this is where you would have to consider the limit from the left and the limit from the right. We can just do that graphically since we have ourselves a nice picture of the graph. So let's take a look at it. So as we're coming towards 2 from the left side, notice how our y's are all approaching that spot where f of 2 equals 3. Coming from the other side, from the right, we're coming down the line this way. But notice we're still approaching the spot where f of 2 equals 3. So in this case, the limit is 3. The third test is that the function value needs to equal the limit, which of course in this case it does. So we conclude that the function f of x is continuous at x equals 2. Notice how specific we are in stating that it is specifically continuous at x equals negative or positive 2. We're not just saying in general, the function is continuous. We're talking about continuity at a particular point. So you do need to make sure you specify at which x value we're talking about in your answer. Notice also the notation, the nice limit notation and the function notation that we have present as we work through the test for continuity.