 So, one of the more important tasks in mathematics is to extend a function beyond an interval where it's known, and this is in general known as the continuation of a function. Now, this is known as a continuation not only because it's an extension, but because generally speaking we want to make sure that the function is extended in a way that it's actually continuous. So, for example, let's say I have a function defined piecewise, and I want to find values C and D that will make this function continuous at x equals 10. So, the first thing we should do is bring in the definition of continuity. So, we need the limit as x approaches 10 of f of x to be L, which means that the limit has to exist. But that requires the limit as x approaches 10 from below and the limit as x approaches 10 from above have to be equal. So, let's find those limits separately. The limit as x approaches 10 from below of f of x. Well, if x is getting close to 10 but staying less than it, that means we're going to be using our first formula, 25x. And so as long as x is slightly less than 10, f of x is 25x, and so our limit is going to be 250. Likewise, as x approaches 10 from above, x is greater than 10, and so f of x looks like our third formula, d minus 10x. And so as x approaches 10 from above, our function approaches d minus 100. And since we want the limit from the left and the limit from the right to be equal, we want 250 to be equal to d minus 100. And so that tells us that d has to be 350. And in that case, the limit as x approaches 10 of f of x has to be 250. And for continuity, we need the limit as x approaches 10 of f of x to be the same as f of 10. And so we need c to be equal to 250. And so there's our value of d and our value of c.