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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. – Alex J Best Aug 6 '12 at 14:17
up vote 11 down vote accepted

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.


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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"Frankly, though, there's nothing wrong with words, especially in something like analysis." - I upvoted for this. – J. M. Aug 6 '12 at 14:11
However, you're missing some incredibly important hypotheses (which first year students of Calculus or even Analysis will often forget). "If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$". Frankly, I don't believe using symbols to express these hypotheses is advisable. – Shaun Ault Aug 6 '12 at 14:18
You don't want to avoid using words, just avoid using them excessively. The same applies for symbols. Writing $\exists$ instead of "there is a..." might be okay when quickly jotting down things for your own use, but otherwise interchanging symbols and words too frequently only makes the text harder to read. Writing using only symbols in a streamlined form can be aesthetically pleasing, but even then some more wordy comments are advisable, as some things are not easily expressible like Shaun mentioned above, and a wall of symbols can easily turn unreadable. – tomasz Aug 6 '12 at 14:20
Words are also symbols, in fact. So the only intention to avoid the use of words is to make the thoughts more smooth. Indeed, if the number of symbols is great enough, it is possible to write an article completely devoid of words, with only the symbols reamaining(I have done something like this.) But imagine what others say, when viewing this... Are you to explain one by one what each symbol means? It takes even more time to do so. As Bourbaki said, the price of the paper is still cheap enough to write things clearly; besides, we are actually typing... – awllower Aug 6 '12 at 14:30
I'd be surprised if $\exists$ foreign students of mathematics at this level who can't understand the English used to state the mean value theorem. – Clive Newstead Aug 6 '12 at 14:53

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