In every instance that I've seen mean value theorem is stated with one condition that I haven't understood, namely the condition of "$f:[a,b] \rightarrow \mathbb{R}$". I know that there are some other important conditions like the function must be continuous on the closed interval $[a,b]$ and it has to be differentiable on the open interval $(a,b)$ but these are easier to observe and give a counterexample for. So why is it important that the function is defined on a closed interval to begin with? Shouldn't it work the same for a polynomial that is continuous and differentiable on $\mathbb{R}$?
Why is it important that we use the Mean value theorem on an interval
mean-value-theoremreal-analysis
Best Answer
The theorem is of course applicable to a function continuous and differentiable on the real line. But the theorem's conclusion needs a particular interval to be fixed in the discussion, to make any sense. That is, the real values $a,b$ must be specified in order to form the concluding statement. (Recall that the theorem's conclusion involves the real numbers $b-a$ and $f(b)-f(a)$.)
The meaning of "$f:[a,b] \rightarrow \mathbb{R}$" is: there is a function taken as given, which assigns to every real $x$ satisfying $a\leqslant x\leqslant b$ a unique real number $f(x).$