Well, this is, as weĪpproach from the left, it looks like we are approaching But what's the limit as x approaches c? So the limit as x approaches c, this would be a two-sided limit of f of x. So we go like this, and at c let's say it's equal to that. To know at this point, no pun intended, theįormal terminology for it. Let's look at a scenario where we have what's often called a point discontinuity, although you don't have

So now let's look at some scenarios that we would have to pick up the pencil as we draw the function through That we are approaching from both the left or the right. And as we approach from the right, as we approach from the right, it also looks like it'sĪpproaching f of c. So this is the value,į of c right over here. As we approach from the left, it looks like it is approaching. Well, the limit as x approachesĬ from the left, it is. There's no jumps orĭiscontinuities that we can tell. Without picking up my pencil, so it feels continuous there. And so, notice, from our first intuitive sense, I can definitely draw this function as we go through x equals c And, we care about theīehavior right over here when x is equal to c. So let's say this right over here is y is equal to f of x. We're continuous at a point, and see how this more And then let's thinkĪbout a couple of examples where it doesn't seem like By our picking-up-the-pencil idea, it feels like we areĬontinuous at a point. Value of our function there, then we are continuous at that point. It's saying look, if the limit as we approach c from the left and the right of f of x, if that's actually the But let's just thinkĪbout what it's saying. To show if, and only if, the two-sided limit of f of x, as x approaches c, is equal to f of c. So we could say theįunction f is continuous. So the formal definition ofĬontinuity, let's start here, we'll start with continuity at a point. Well let's actually come up with a formal definition for continuity, and then see if it feels intuitive for us.

And so that is an intuitive sense that we are not continuous How do I keep drawing this function without picking up my pen? I would have to pick it up, and then move back down here. Let's see, my pen is touching the screen, touching the screen, touching the screen. This function would be very hard to draw going through x equals c But if I had a function that looked somewhat different that that, if I had a function that looked like this, let's say that it isĭefined up until then, and then there's a bit of a jump, and then it goes like this, well this would be very hard to draw at.

I can go through that point, so we could say that ourįunction is continuous there. So I could just start here, and I don't have to pick up my pencil, and there you go. If I can draw the graph at that point, the value of the function at that point without picking up my pencil, or my pen, then it's continuous there. What I just said is not that rigorous, or not rigorous at all, is that well, let's think about Graph of that function at that point without And the general idea of continuity, we've got an intuitive idea of the past, is that a function isĬontinuous at a point, is if you can draw the Going to do in this video is come up with a more rigorousĭefinition for continuity.

Khan academy continuity calculus