 So one of the more important ideas in mathematics is known as continuity, and for many years mathematicians treat it continuity in the same way that we treat a number of other things, we know it when we see it. However, this is not good enough for formal mathematics, because it means that what one person sees might not actually be seen by somebody else, so we want some sort of useful definition, and so we define continuity in the following way. I'm going to take a function f of x, and it's going to be continuous at a particular point, if and only if, the function value and the limit of the function as x gets close to a are both existent quantities and are equal to each other. Now, in the course of the semester, we will probably be asked to prove one thing or another, and you might be asked to prove that a function is continuous at a point or over some interval. Now, a very important thing to keep in mind is that the only acceptable answer to a question like prove that a function is continuous is going to make some reference back to this definition of continuity. There are very few things you will be asked to memorize in mathematics, but the definitions are the absolute minimum. If you do not know the definitions, you cannot proceed. In this particular case, the definition of continuity has three important components. First off, we need to find f of a, make sure that it exists. We need to find the limit as x approaches a of f of a, and then we want to show that the limit and the function value are equal, in which case the function will be continuous, or possibly the limit and the function value are different, in which case the function will be discontinuous at the point in question. And just to make the problems interesting, we won't commit ourselves in the question to whether or not the function is or is not continuous. Prove or disprove f of x equals one over x is continuous at x equals zero. So maybe the function is continuous, maybe it's not. Well, let's take a look at it. So in order to prove continuity, we need to make sure that the function value at a point and the limit as x approaches that point are both existent, they both exist, and they are equal. So generally speaking, it's easier to start by finding the function value, so we might try and find the function value at x equal to a. So our point of interest x equals zero, our definition x equal to a, so that says that zero and a are the same thing, so that a is equal to zero, so we can start by finding f of zero. Well, let's go ahead and find that out. f of x equals one over x, so f of zero is going to be one over zero, and we have a problem because we cannot put a zero in the denominator. So that tells me f of zero is undefined. Well, there goes our any possibility of continuity because f of a has to exist. f of zero does not exist, so f of x cannot be continuous at x equal to zero. So we should say so, and then as the complete answer to our question prove or disprove f of x equals one over x is continuous at x equals zero, we want to include the portion that's in green. The function value is undefined, so that means that our function is not continuous. It's worth pointing out that while you should know the definition of continuity, you don't actually quote it at any point in the course of the proof. What you should have is what's called a working understanding of the definition of continuity, and in this particular case that working concept of the definition of continuity means that you know that continuity requires the function value exist, and as soon as the function value fails to exist, then you can conclude the function is not continuous. Well, let's take a look at another example, prove or disprove f of x equals the cube root of x is continuous at x equal to eight. So again, we want to find the function value at eight, so we'll just substitute eight into our function formula, f of eight equals the cube root of eight, and resolve that f of eight equals two. The next thing we want to do is you want to find the limit as x approaches eight of our function. So limit is x approaches eight of f of x, it's the limit is x approaches eight of the cube root of x, and as x gets close to eight, cube root of eight gets close to two. And so there's my limit value, and so far so good, the function exists, the limit exists, the requirement that we have for continuity is not only that they both exist, but that they are equal. So we want to compare them, and they are exactly the same. So because my function and my limit values are equal, then the function is continuous at x equals eight, and my complete answer to this question is the portion shown in green. I have the function value, I have the limit value, they're equal, so I can make the conclusion that the function is continuous.