I've been working hours on translating this one statement into formal mathematical language:
- If $\textit{f}$ is a continuous on the interval $[\textit{a},\textit{b}]$ and differentiable on $(\textit{a},\textit{b})$, then there is a number $\textit{c}$ $\in$ $(\textit{a},\textit{b})$ for which $\textit{f}^\prime(\textit{c}) = \tfrac{f(b)-f(a)}{b-a}$.
What I have come up with are (these are my best versions):
- Informally,
$\forall$ func. $f : \mathbb{R} \rightarrow \mathbb{R}$, ($f$ continuous and differentiable) iff $\exists$ $a,b$ $\in$ $\mathbb{R}$ s.t. $f$ continuous on $[a,b]$ and $f$ differentiable on $(a,b)$ $\implies$ $\exists c \in (a,b)$ s.t. $\textit{f}^\prime(\textit{c}) = \tfrac{f(b)-f(a)}{b-a}$.
- Formally,
($\forall f$)(
[($f: \mathbb{R} \rightarrow \mathbb{R}$) & (($f$ cont.) & ($f$ diff.))] $\iff$ (
($\exists a,b$)([$a,b \in \mathbb{R}$] & [[($f$ cont. on $[a,b]$) & ($f$ diff. on $(a,b)$)] $\implies$ [($\exists c)([c \in (a,b)$] & ($\textit{f}^\prime(\textit{c}) = \tfrac{f(b)-f(a)}{b-a}$))]
]))).
I'm not at all confident with what I'm doing.