 Hello, and welcome to this screencast on section 10.1, Limits. This screencast is going to cover continuity. Let's first review continuity for single variable functions. Recall that a function f of a single variable is said to be continuous at x equals a, provided that the following three conditions are satisfied. First, that f of a exists. Second, the limit of f as x approaches a exists. And third, that these expressions are equal. Using our understanding of limits of multivariable functions, we can define continuity for two variable functions in the exact same way. Just as with single variable functions, continuity has certain properties that are based on the properties of limits. We can use these properties along with what we know from single variable calculus to show that other functions are continuous. For example, consider this function f. First, since x is continuous everywhere by property one from the previous slide, we know that 5x is also continuous everywhere. Since y cubed is continuous everywhere, then by property four, 5 times x times y cubed is continuous everywhere. Since x squared is continuous everywhere, by property two, 5xy cubed plus x squared is continuous everywhere. Therefore, we have shown that the function f is continuous everywhere.