On Wikipedia, when Proving a generalization of the mean value theorem, they state "Suppose f : [a, b] → R is continuous and g is a nonnegative integrable function on [a, b]. By the extreme value theorem, there exists m and M such that for each x in [a, b], $ {\displaystyle m\leqslant f(x)\leqslant M} $and $ {\displaystyle f[a,b]=[m,M]}$. They then proceed to make several evaluations. The part I am confused about is the final statement: $${\displaystyle f[a,b]=[m,M]}$$ What is this intended to mean( it was pointed out what I previously said here was incorrect, and I couldn't find the strike option, so I removed and replaced it). I couldn't find anything obvious by searching google, so I Decided to ask here. If this is the wrong place to ask this, please move it to the right place. Thanks!
( The proof is located at https://en.m.wikipedia.org/wiki/Mean_value_theorem under MVT for definite integrals, the first proof )
It means that the image of the interval $[a, b]$ under the function $f$ is the interval $[m, M]$. I.e., for every $x \in [a, b]$, $f(x) \in [m, M]$ and for every $y \in [m, M]$ there is an $x \in [a, b]$ such that $f(x) = y$. If $f$ is a function and $X$ is a set it is standard to use $f(X)$ to mean image of $X$ under $f$, i.e., the set $\{y \mid \exists x \in X(f(x) = y)\}$. The Wikipedia article is using something like this notation but writing $f[a, b]$ rather than $f([a, b])$. (Many writers, myself included, prefer to write $f[X]$ for the image, so we'd write $f[[a, b]]$ in this case.)