Consider the integral $\int_a^b f(t)dt$.
There are many numerical integration methods, like trapezoidal rule, Simpson's rule, Gaussian quadrature, but all they involve derivative in the error bound.
For example, error for the Simpson's rule is $\frac{(b-a)^5}{180N^4} \operatorname{max}_{u\in[a,b]}|f^{(4)}(u)|$.
But what if $f(t)$ is nowhere differentiable? What numerical method should I use?
Are there numerical methods, that don't involve derivative in the error bound?
If $f$ is of bounded variation $V(f)$, then Koksma's inequality (Theorem 5.1 in Kuipers & Niederreiter, Uniform Distribution of Sequences) says $$ \left|{1\over N}\sum_{n=1}^Nf(x_n)-\int_0^1f(t)\,dt\right|\le V(f)D_n^* $$ where $D_N^*$ is the discrepancy of the point set $\{\,x_1,x_2,\dotsc,x_n\}$.
Theorem 5.4 in the same book says that if $f$ is continuous and has modulus of continuity $M$, then $$ \left|{1\over N}\sum_{n=1}^Nf(x_n)-\int_0^1f(t)\,dt\right|\le M(D^*_N) $$ There are other formulas in the book for estimating integrals.
$\newcommand\Z{\mathbb Z}\newcommand\R{\mathbb R}\newcommand\la{\lambda}$If $f$ is an arbitrary Lebesgue-integrable function, then, as is done in a definition of the Lebesgue integral, it makes sense to group together close values of the function $f$, rather than close values of its argument.
So, say, for any Lebesgue-integrable function $f\colon[0,1]\to\R$, any real $a$, and any real $h>0$, $$\Big|\int_0^1 f(x)\,dx-\sum_{k\in\Z}(a+k+1/2)h\,\la(E_{f,a,h,k})\Big|\le\frac h2,$$ where $E_{f,a,h,k}:=f^{-1}((a+kh,a+(k+1)h])$ and $\la$ is the Lebesgue measure.
Of course, if $f$ is monotonic, then the sets $E_{f,a,h,k}$ will be intervals. In this case, the bound $\frac h2$ will be just as effective as the Koksma bound $V(f)D_n^*$ in Gerry Myerson's answer (provided the set $\{x_1,\dots,x_n\}$ is of the minimal discrepancy, $\frac1{2n}$ -- see Theorem 1.4 on p. 91).
If $f$ is not monotonic (think e.g. of $f(x)=\sin mx$ for large $m$), then the bound $\frac h2$ can be much better than the Koksma bound $V(f)D_n^*$. As for the calculation of the variation $V(f)$ (when it is finite, even when $f$ is not monotonic), it seems to be of about the same complexity as the calculation of the $\la(E_{f,a,h,k})$'s.
Also, it seems that the bound $\frac h2$ will be more effective, or significantly more effective, than the bound $M(D_n^*)$ in Gerry Myerson's answer, even when the set $\{x_1,\dots,x_n\}$ is of the minimal discrepancy. For instance, when $f(x)=x^p$ for a real $p>0$ and all $x\in[0,1]$, $a=0$, and $h=1/n$, then the bound $\frac h2$ is $\frac1{2n}$, whereas the bound $M(D_n^*)$ is $\sim \frac p{2n}$ if $p\ge1$ and $=\frac1{(2n)^p}$ if $p\in(0,1]$ (again, provided the set $\{x_1,\dots,x_n\}$ is of the minimal discrepancy, $\frac1{2n}$).
