I'm not going to prove it here, but I think the conceptual underpinning here should be straightforward. So the theorem tells us that suppose f is a function continuous at every point of the interval, the closed interval, so we're including a and b, so it's continuous at every point of the interval a, b. And so let me just draw a couple of examples of what f could look like just based on these first lines. So suppose f is a function continuous at every point of the interval a, b. So let me draw some axes here. So that's my y-axis, and this is my x-axis. So one situation, if this is a, and this is b, f is continuous at every point of the closed interval a and b, so that means it's got to be, for sure, defined at every point as well as being continuous.