I'm looking for a stringent definition of the mean value theorem i.e. stated with mathematical symbols. I think it should be something like "there exists..." and then the mean value if there is an integral between the surrounding values which to my knowledge is the mean value theorem, stating that the mean has to be somewhere between two points. So could we define it in mathematical notation?
" there exists a point c in $(a, b)$ such that $f'(c)= \frac{f(b)-f(a)}{b-a}$ "
Can we rephrase the above into a "pure" mathematical notation i.e. using ∃ instead of plain words? Is it trivial what I want to do and just replace the "there exists" with ∃ so that it is more like a logical statement?