 In this video, I want to demonstrate how to show and determine at which points a piecewise function is continuous and by contrast, where is a piecewise function discontinuous. And so by example, consider the function f of x as illustrated here, it's going to be x plus one when x is less than one. It'll be x squared minus three x plus four when x is between one and three inclusive, and it'll look like x minus or five minus x when x is greater than three. All right, so this function right here, it's a piecewise function, but notice what the pieces look like. We have a polynomial, we have a polynomial, we have a polynomial. This piecewise function was built together using polynomials, which as we've learned previously polynomials are continuous functions. So this is what we often call a piecewise continuous function. That is, it's a function piece together using continuous functions. The advantage of a piecewise continuous function is that when you're inside a piece, you're going to be continuous. Like if you look at the piece x, y equals x plus one, you see this line right here, everywhere in the middle of the piece, you're going to be continuous at those, at those points. Likewise, if you look at the parabola x squared minus three x plus four, everywhere on the interior of that interval, you're going to be continuous. And lastly, if you look at the piece that corresponds to y equals five minus x, everywhere in the interior of the piece, you're going to be continuous. You're continuous at all those places. What you're lacking potentially is going to be continuity at the switching numbers. Look at these numbers one and three where it switches between pieces. A function we say is continuous. If you could draw that function, you know, essentially with one continuous stroke of your pen, right? If you can draw the whole picture without having to pick up your pen, that means your function is continuous. Even if you have to stop and pause for a second, right? Like the absolute value function is an example of a continuous function. It's okay that if you come and stop and then switch directions, the point is you don't have to pick up your pen to draw the absolute value function. So piecewise continuous functions will be continuous at their switching numbers if they'll be continuous at the switching numbers if they touch each other on the graph. So when we look at this thing graphically, we can see that at x equals one, it'll be continuous because the two pieces touch. But at x equals three, it'll be discontinuous. There'll be a jump discontinuity here. So you're continuous at x equals one, but you're going to have a jump discontinuity at x equals three. We can see that from the graph. How do we detect that algebraically? Well, when you have a piecewise continuous function, you're basically the only thing you have to look out for is jump discontinuities. Removable discontinues are also a possibility if you kind of just moved a point, but these jump discontinues are really the concern. So you're going to be looking at these switching numbers. So you're going to want to consider what is the limit as x approaches one from the left of f of x and what is the limit as x approaches one from the right of f of x. Now with a piecewise function, if you're a little bit to the left of one, you're in this interval right here, so your function is going to look like x plus one. So we're looking at the limit as x approaches one from the left of x plus one. But as x plus one is a polynomial to continuous function, you can evaluate the function to compute the limit. So this will look like one plus one, which is equal to two. On the other hand, if we're taking the right-handed limit of one of f of x, if you're a little bit bigger than one, then you're going to be in this interval right here. And so the function will look like the parabola. It'll look like x squared minus three x plus four as x approaches one here from the right. In this case, this polynomial is also a continuous function. We can evaluate the function to find the limit. So we're going to end up with one squared minus three times one plus four. And so we end up with one minus three plus four. One minus three is negative two plus four is equal to two. So therefore, because the left-handed limit was two and the right-handed limit was two, this means that the limit exists. The limit as x approaches one of f of x, this equals two, right? So the limit exists here. It's going to be continuous at x equals one. Now, we see a different thing happening when we look at x equals three. If we look at the left-handed limit as x approaches three for f of x, this again would be in the domain that gives us the parabola. So this limit as x approaches three from the left will be of the parabola x squared minus three x plus four. If we evaluate this at three, we're going to get three squared minus three times three plus four. You're going to notice here that three squared is nine. You also have a negative nine there. Those are going to cancel with each other. You end up with the value four, okay? That's the left-handed limit. On the other hand, the right-handed limit, if we compute that one, the limit as x approaches three from the right of f of x. If you're a little bit bigger than three, you're going to fall inside this domain. So the function will look like the line five minus x as x approaches three. But as this is a continuous piece, to evaluate, we just plug in three. So we get five minus three, which gives us two. And you'll see that in this situation that two does not equal four. So the left limit and the right limit don't agree with each other. The limit doesn't exist. The function is discontinuous. And so when you have a piecewise continuous function, the only numbers you have to worry about discontinuities will be at the switching numbers. If the left limit and the right limit agree with each other, then the function will be continuous. If the left limit and the right limit disagree with each other, then it'll be discontinuous. But computing the left limit and the right limit will just be evaluating the function at that port associated to the left of the switch and to the right of the switch. And so you just have to make sure the two functions touch each other when you switch pieces. Thank you.