 One of the most important concepts in calculus is the concept of continuity. The problem is continuity is a geometric concept. We know it when we see it. And so we know what a continuous graph looks like. And we know what a not-continuous graph looks like. But the problem is calculus is an algebraic concept. We have formulas. And so the question is how do I translate these geometric concepts into an algebraic framework? And so we'll run a little bit of analysis here. So I suppose my graph is continuous at some point where x is equal to a. Then I notice that there's two important things that happen. First of all, there's actually a point on the graph a f of a. Moreover, as we get close to x equal to a, our y values get close to f of a. And since y equals f of x, this means that the limit as x approaches a is f of a. On the other hand, if y equals g of x is not continuous at x equals a, we see one of several things. First, there might not be a point on the graph at x equals a. g of a might not exist. Next, there might not be a value for the limit as x approaches a of g of x. And even if there is a point, a g of a, and the limit exists, they might not agree. And this suggests the following definition for continuity. A function f is continuous at x equals a, if and only if f of a is equal to some value l. In other words, f of a itself exists. And the limit as x approaches a of f of x is equal to that value l. We can extend this and say the function is continuous over some interval between a and b, if and only if it is continuous for all x in that interval. For example, suppose f has the following function values. Let's find a value c that will make f of x continuous at x equals 2. So let's pull in our definition of continuity. And if we want the function to be continuous, we need the function value to be equal to the limit. So the first question we want to ask ourselves is, self, what's the limit as x approaches 2 of f of x? And the table suggests that the limit as x approaches 2 of f of x is equal to 9. So if our function value is equal to 9, then our limit as x approaches 2 and our function at 2 are the same. And so our function will be continuous at x equals 2. Now previously we had established a theorem about the limits of algebraic functions. And this theorem, combined with our definition of continuity, leads to an extremely important result. Let f of x be any function expressed in terms of powers, roots, and or quotients of our variable x. Then f of x is continuous everywhere it's defined. And that's because the limit as x approaches a of f of x will be f of a, according to our limit theorem, and because the limit is the function value, then the function will be continuous according to our definition of continuity. We can actually go a little bit farther with this. Algebraic functions are those that can be expressed using a finite number of powers, roots, and or quotients of the variable x. So algebraic functions are friendly things like f of x equals xq plus 5x plus 7, or fifth root of xq plus 7 over 2x squared minus 7x minus 1, or some horrible expression like this. On the other hand, transcendental functions cannot be expressed using finite numbers of powers, roots, and or quotients of our variable. And as a result, transcendental functions look like these horrifying expressions like 5 to the x, or log of 3x plus 5, or sine of 3x. And as you can probably guess from these examples, our primary transcendental functions are the exponential, logarithmic, and trigonometric functions. And rather conveniently, it turns out that what is true for algebraic functions also turns out to be true for transcendental functions. And so we have a theorem. Algebraic and transcendental functions are continuous everywhere they are defined.