 Let's look at a couple of other examples of how to determine continuity of a function at a specified point. Once again, we're going to take this from two different perspectives, from a graphical perspective, and also from utilizing the three tests for continuity at a point that we have in calculus. So both of these examples are going to be piece functions. And if you consider this first one, really what's happening here is it's essentially a linear equation, but the whole that would exist from that first piece at x equals 3 gets filled in by the second piece. So let's take a look at it graphically first. So if you consider that first piece 10 minus 2x, of course that's going to be a linear equation crossing the y-intercept at 10 and having a slope of negative 2. So let's start by graphing that. So here's 10 and our slope is negative 2. So we'll come down 2 over 1, down 2 over 1, down 2 over 1. Now this is where x equals 3 on this next one. That's where we're going to have a hole. So right here we would have a hole and we keep going down 2 over 1, so it looks like this. That's from the first piece. The second piece, though, states that when x equals 3, your y value is given by x plus 1. And notice that then fills in the hole. So really all you have in the end is a line. So let's take a look at this and how it would work out with the tests for continuity. Remember the first test is that the function value needs to exist. Well in this case the function value of 3 is given by that second piece. So we know that to be 4, 3 plus 1. Second step is that the limit as we approach 3 of this function has to exist. Of course we do need to consider both the left and right hand limits. If you wanted to do this graphically you could. There's 4 and you can see that the graph is approaching the y value of 4 as you approach the x value of 3 both from the left and from the right. If you wanted to do it algebraically remember you'd be substituting into the first piece because if you're thinking of values a little less than 3 and a little bit bigger than 3 the first piece applies in each of those cases. So it's into that first piece that we would substitute 3. Remember you actually substitute that value in itself and you do get 4. The third test of course then is that the function value has to equal that limit value which it does in this case. So we would conclude that this function f of x is continuous at x equals 3. So let's look at another piece function. Let's start out by graphing it and checking it out that way. Well 1 over x we know what that looks like. That's just going to look like two branches. One in the first quadrant, one in the third quadrant. But where x equals 0 though we have a y value of 5. Suppose that's 5 up there. So you can tell it's not continuous. It's an infinite discontinuity and there's just this random point on the y axis. So let's check it out from the perspective of using the test for continuity. So first we need to determine if f of 5 exists. Well it does and we're told it equals 0. Now we need to consider the limit of the function as we approach 0. Well this is where obviously it does not exist. The limit as you approach 0 from the left does not exist but approaches negative infinity. Coming from the right side of 0 the limit does not exist but approaches positive infinity. So this is where that one falls apart is at the limit step. So f of x therefore is not continuous at x equals 0.