1. ## Approximate Fundamental Theorem

I am looking for an approximate form of the Fundamental Theorem that looks like: let $f$ be $C^1$ on $[a,b]$; for every $\epsilon > 0$, whenever $P$ is a partition of $[a,b]$ s.t. $||P|| < \cdots$ (some term depending on $\epsilon$), $|\int_a^bf'(x)dx - (f(b)-f(a))| < \epsilon$.

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Let $P$ be a partition of $[a,b]$, $P : a=x_0 \leq x_1^* \leq x_1 \leq x_2^* \leq \cdots \leq x_n = b$, and write $||P|| := \max\ (x_i-x_{i-1})_{i=1}^n$ for its mesh size. $\sum_P f := \sum_{i=1}^n f(x_i^*) (x_i - x_{i-1})$ denotes a Riemann sum of $f$ on this partition.

Now:

$\left| \int_a^b f'(x) dx - \left( f(b) - f(a) \right) \right| \leq \left| \int_a^b f'(x) dx - \sum_P f' \right| + \left| \sum_P f' - \left( f(b)-f(a) \right) \right|$

The first term can be made $\leq \epsilon$ by chusing $P$ fine enough (this is the definition of integrability), so I ignore it and look at the second term. I can make it vanish by chusing the "right" $x_i^*$ but suppose I cannot chuse these and can control only the mesh size of $P$.

Using the formula $f(x_i)-f(x_{i-1})=f'(x_i^*)(x_i - x_{i-1}) + \mathrm{error}(x_i^*;x_i-x_{i-1})$ where the error term is $o(x_i-x_{i-1})$ as $x_i - x_{i-1} \to 0$...

$\sum_P f' - \left( f(b)-f(a) \right)$
$=\sum_{i=1}^n f'(x_i^*) (x_i - x_{i-1}) - \sum_{i=1}^n \left( f(x_i) - f(x_{i-1}) \right)$
$=\sum_{i=1}^n \mathrm{error}(x_i^*;x_i-x_{i-1})$

I want to show this last sum can be made arbitrarily small by shrinking $||P||$. I do not know how to control the error terms. Can anyone help?