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The conditions for the Mean value theorem is that if $f$ is defined on a closed interval $[a,b]$, f is continuous on [a,b] and differentiable on $(a,b).$ Then there exists a $\xi$ in [a,b] such that $f'(\xi)=\frac{f(b)-f(a)}{b-a}$ I want to express this in mathematical notation. Like this:

$f:[a,b]\rightarrow \mathbb{R}$ , $f\in C[a,b], f\in C^{1}(a,b)$

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

Is this correct?

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  • $\begingroup$ $f$ being $C^1$ on $(a,b)$ is too strong because that would mean its derivative is continuous, which you don't need. Otherwise, I think what you wrote is okay. $\endgroup$ – Max Sherman Nov 18 '13 at 23:06
  • $\begingroup$ If $f$ is differentiable in $(a,b)$, then it follows that it is continuous in $(a,b)$? How do you write that in symbolical notations otherwise? $\endgroup$ – EricAm Nov 18 '13 at 23:10
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    $\begingroup$ I am taking $f \in C^1$ to be a function that is continuously differentiable, meaning that in addition to being continuous and differentiable, $f$ also has a continuous derivative. The mean value theorem doesn't need the derivative of $f$ to be continuous. That's all I'm saying. $\endgroup$ – Max Sherman Nov 19 '13 at 2:53
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Yes, this is correct. You could also say $f \in C[a,b] \cap C^1(a,b)$. Though I should note that while this is convenient for your notes, or maybe for writing on a chalkboard, in your formal mathematical writing, you should aim to be a little bit less terse. In particular, never start a sentence with a symbol, and when writing paragraphs, avoid using symbols for logical terms like $\Rightarrow, \exists, \forall$ and shorthand like "w.r.t.", "s.t." or ":" (there are exceptions, like $\{ x : x > 2\}$). So something like:

Let $f : [a,b] \to \mathbb R$ such that $f \in C[a,b] \cap C^1(a,b)$. Then, there exists $\xi \in (a,b)$ such that $f'(\xi) = \dfrac{f(b) - f(a)}{b-a}$.

would be appropriate if you're writing up your homework, or a research paper or something.

Here's a good guide on style: http://www.math.washington.edu/~lee/Writing/writing-proofs.pdf

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