The limit as x approaches c of f of x, so let's say that f of x as x approaches c is approaching some value. So if we approach from the left, we're getting to this value. If we approach from the right, we're getting this value. Well, in order for the function to be continuous, if I had to draw this function without picking up my pen, well, the value of the function at that point should be the same as the limit. This is really just a more rigorous way of describing this notion of not having to pick up your pencil, this notion of connectedness, that you don't have any jumps or any discontinuities of any kind. So with that out of the way, let's discuss continuity over intervals. Let me delete this really fast so I have space to work with.