What is the intermediate value principle of continuous function?

What is the intermediate value principle of continuous function?

Let y = f (x) be continuous on the closed interval [a, b], then the values at the end of the interval are different, that is, f (a) = a, f (b) = B, and a ≠ B. then, no matter what number C is between a and B, there is at least one point ξ in the open interval (a, b), such that f (ξ) = C (a)

How to prove the boundedness of continuous functions on closed intervals by using the finite covering theorem

Because it is continuous, every point has a limit, so we can find an open interval, so we have an open cover, so we have a finite number, so we are bounded

What is the boundedness theorem of continuous function

A continuous function on a closed interval is bounded on the interval and obtains its maximum and minimum values ～