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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?

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    IMO the only thing lacking in your statement in terms of a stringent definition is your assumptions. As mentioned below words are great, if the meaning is clear and there are no ambiguities, so phrases like "there exists" are fine.2012-08-06

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You could write that as

$\exists c \in (a,b)\, :\, f'(c) = \frac{f(b)-f(a)}{b-a}$

Some people might prefer to use other conventions, such as $\exists c.\ c \in (a,b) \wedge f'(c) = \frac{f(b)-f(a)}{b-a}$

Frankly, though, there's nothing wrong with words, especially in something like analysis. It doesn't make it any less rigorous as long as the language used isn't ambiguous.


Edit:

Amused by one of the comments, I thought I'd write out in full the statement of the mean value theorem using only symbols.

$(\forall a,b \in \mathbb{R})(\forall f \in \mathbb{R}^{[a,b]})(((\forall x \in [a,b])(\forall \epsilon > 0)(\exists \delta > 0)$ $(\left|x-y\right|<\delta \Rightarrow \left|f(x)-f(y)\right|<\varepsilon)) \wedge ((\forall x \in (a,b))(\exists f'(x) \in \mathbb{R})$ $(\forall \varepsilon > 0)(\exists \delta > 0)(\forall h>0)((h<\delta) \Rightarrow \left| \frac{f(x+h)-f(x)}{h} - f'(x)\right| < \varepsilon))$ $\Rightarrow (\exists c \in (a,b))(f'(c) = \frac{f(b)-f(a)}{b-a}))$

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    Is this a both ways relation? If no f(c)=0 exists then f(a)!=f(b) ?2017-05-03