 So let's check a couple of functions for continuity. So let f of x be some function defined piecewise, and note that we have three different rules depending on where we are. If x is less than 5, we use the rule 25 plus x. If x equals 5, we use the rule 10, and if x is greater than 5, we use the rule x squared. For x less than 5, our function is 25 plus x, which is algebraic. And one of the things we know is that algebraic functions are continuous everywhere, so f of x is continuous for x less than 5. Likewise, for x greater than 5, f of x is x squared, which is also algebraic, and so our function is also continuous for x greater than 5. At x equals 5, we change rules, and so we need to check for continuity. So returning to our definition of continuity, remember that a function is continuous at x equals a, if and only if the function value is equal to something, and the limit as x approaches a of the function is equal to the same thing. So the easy thing to check first is the function value, and we find that f of 5 is equal to 10, so f of 5 exists. We also need to check the limit as x approaches 5 and f of x. Since the rule changes at x equals 5, we'll approach from below and from above. So as x approaches 5 from below, f of x is the function 25 plus x, and since this is algebraic, we'll just let x be 5 and we'll get a limit value of 30. As we approach 5 from above, our function is x squared, and so our limit as x approaches 5 from above is going to be the limit as x approaches 5 from above of x squared, which will be 25. Since the limit as x approaches 5 from below is not equal to the limit as x approaches 5 from above, then the limit as x approaches 5 does not exist. So this means that f of x is not continuous at x equals 5. Or we might take another function, also defined piecewise, and to begin with we note that since g is algebraic, it will be continuous for x less than 10 and for x greater than 10. Going over our definition of continuity, we need to verify that g of 10 exists, and we do have a definition if x equals 10, g of x is going to be 50, so g of 10 exists, and we'll check the limit. And again, because our rule changes at x equals 10, we'll have to approach from both sides. As x approaches 10 from below, our function is x squared, and our limit is going to be 100. As x approaches 10 from above, our function is 20x minus 100, and our limit is 100. And since the limits agree, we can say the limit is 100. So the limit exists, but since the limit is not equal to the function value, the function is not continuous at x equals 10.