Differentiability

If $f: [a, b] \to {\mathbb{R}}$ is continuous on a closed interval and differentiable on $(a, b)$, then there exists $\xi \in [a, b]$ such that $$\begin{align*} f(b) - f(a) = f'(\xi)(b-a) .\end{align*}$$

More generally, if $g: [a,b]\to {\mathbb{R}}$ is similarly continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists a $\xi$ with $$\begin{align*} \qty{ f(b) - f(c) } g'(\xi) = \qty{g(b) - g(a)} f'(\xi) .\end{align*}$$ What this means graphically:

If $\left\{{f_n}\right\}$ is a sequence of functions where

• each $f_n$ is differentiable,
• there is some $G$ such that ${\left\lVert { \sum_{n\leq N} f_n' - G} \right\rVert}_\infty \overset{N\to\infty}\longrightarrow 0$, and
• there exists at least one point ¹ $x_0$ such that $\sum f_n(x)$ converges (pointwise),

then there exists an $F$ such that ² $$\begin{align*} {\left\lVert { \sum_{n\leq N} f_n - F} \right\rVert}_\infty \overset{N\to\infty}\longrightarrow 0 && F' = g .\end{align*}$$