Theorems covering limits and continuity of functions
Depending on the teacher and on which type of calculus course it is being taught, those theorems may be left unproven because they require a level of abstraction that not everyone is familiar with. There is also a matter of time constrains. It's not always feasible to do the proofs.
The theorems below always assume functions to be continuous, because if they aren't continuous in some interval we are unable to state many properties. There are always two different points [math]\displaystyle{ a }[/math] and [math]\displaystyle{ b }[/math], because if [math]\displaystyle{ a = b }[/math] we don't have an interval and cannot state anything about the function except for checking whether the function is defined there or not. The distance between the extreme points can be small or large. As long it's positive it doesn't matter. The graphs may differ from textbook to textbook and that's natural. We are taking one small part of a general idea. There is no way to plot all cases one by one, nor plot at infinity.
Before progressing to the theorems we can already make some questions: what if the function has a point in which the limit is infinity? Then the function cannot be continuous at that point. Therefore, we can't have a maximum value there because infinity is not a number. Conversely, negative infinity is not a minimum value. What if the limit doesn't exist? At best we can state that the function is bounded between two extremes, but with the limit being undefined we are unable to conclude anything from a value that we can't know for sure. Can you answer the question of which number is greater than the other if you don't know the numbers in the first place? It's impossible! By assuming the functions to be continuous in a certain interval we are ruling out some cases which would invalidate the theorems otherwise.
Weierstrass' extreme value theorem
If [math]\displaystyle{ f }[/math] is continuous in [math]\displaystyle{ [a, \ b] }[/math]. Then there shall exist [math]\displaystyle{ x_1 }[/math] and [math]\displaystyle{ x_2 }[/math], in that interval, such that [math]\displaystyle{ f(x_1) \leq f(x) \leq f(x_2) }[/math] for all [math]\displaystyle{ x }[/math] in [math]\displaystyle{ [a, \ b] }[/math].
(the theorem does not care about the function being constant, crescent or decrescent in between the two extremes. Don't associate the theorem to just one type of function.)
What this theorem states is that, in between a closed interval, the function is going have a maximum and a minimum. The interval has to be closed because if we consider an open interval we are accepting values at which the limit may be infinity, which would invalidate the theorem. Remember that infinity is not part of the real numbers. We cannot calculate a function at infinity, but we can calculate the limit at infinity. By having the interval closed we guarantee that the function is defined and has a finite limit.
Suppose we have the function [math]\displaystyle{ f(x) = 1/x }[/math]. If the interval is open and contains the zero, we won't ever divide by zero. However, by being open we can get infinitely close to zero and this means to accept that the function is not bounded because it's going to infinity. If the interval is closed and excludes the zero, maybe it has the number 0.0001 which is finite and [math]\displaystyle{ f(0.0001) }[/math] is finite. Anything in between 0.0001 and 0 is not part of the interval.
Rolle's theorem
If [math]\displaystyle{ f }[/math] is continuous in [math]\displaystyle{ [a, \ b] }[/math], differentiable in [math]\displaystyle{ ]a, \ b[ }[/math] and [math]\displaystyle{ f(a) = f(b) }[/math]. Then there shall exist a [math]\displaystyle{ x_1 }[/math], in that interval, such that [math]\displaystyle{ f'(x_1) = 0 }[/math].
(the theorem guarantees that at least one point is going to have a derivative equal to zero. There may be more than one.)
What this theorem states is that in between two points of the function with the same height, there is going to be a point where the derivative is zero. Think about this: if the function is strictly crescent or strictly decrescent it's impossible for it to have an horizontal tangent in between two points. The theorem states that if the two extremes have the same value, either the function is constant or somewhere in between the rate of change invert its sign.
For the same reason of the extreme value theorem the interval has to be closed. Now why does it say that the function is differentiable on an open interval? Because there are functions with points where the limit exists and the function is also continuous, yet it's not differentiable. For example [math]\displaystyle{ f(x) = |x| }[/math] is differentiable everywhere, except for the origin. That's why the theorem says that the function is differentiable on an open interval. The function may be continuous at the extremes, while at the same time not differentiable there.
Mean value theorem
What this theorem states is that we have a secant line that passes through two points of the function. In between the two points there must be a tangent line that is parallel to the secant.
