A Bound for the Error of the Numerical Approximation of a the Integral of a Continuous Function How to numerically integrate a nasty function? 
Suppose $f$ is only continuos; which method can you employ to approximate 
$$\int_0^t f(s)ds$$
Since $f$ is continuos the integral exists, but all the numerical approximation methods I studied bound the error term with the hypothesis that $f$ is at least $C^2$ or something. 
I also know of the left rectangle method that only requires $f$ to be holder-continuos for some $\alpha$, but suppose this $f$ is not even Holder continuos.
Can we find a meaningful error bound? 
 A: This is where using randomness can help. Let $X_i\sim U(0,t)$ be independent and let $$I_n=\frac t n\sum_{i=1}^n f(X_i)$$
Then $I_n$ converges to $I=\int_0^tf(x)\,dx$ a.s. and in $L^2$: $$\sigma^2_n=E((I_n-I)^2)=\frac {\int_0^tf^2(x)\,dx-I^2}n\to 0$$ regardless of how hairy your function is.
To get the simplest bound it's enough to know $M=\max_{x\in[0,t]} |f(x)|$ for then $\sigma_n^2\le\frac {M^2t}n$.
A: 
Can we find a meaningful error bound? 

No. At least, using conventional terms.
Consider $h(x) = \begin{cases} d-d^2*|x-x_0| & | x \in [x_0-{1 \over d}, x_0+{1 \over d}] \\ 0 & otherwise \end{cases}$. When $[x_0-{1 \over d}, x_0+{1 \over d}] \subseteq [0,t]$, $\int_0^th(s)ds = 1$. But unless we somehow try to compute value of h(x) on arbitrarily narrow interval, $h(x)$ can't be distinguished from $0$.
Or, more strictly, let's suppose we have such estimation for given numerical method. Let's take all iterations until error is less than 1. Then let's apply the same method for $f_1(x)=f(x)+2h(x)$, where initial method application didn't estimate $f$ anywhere on $[x_0-{1 \over d}, x_0+{1 \over d}]$. You will have the same result with error less than 1, but actual results will differ by 2.
In more general case, we can take $f_1(x)$ which equals $f(x)$ in all points where our method estimated it and still have $f_1(x)=C$ for any $C$ and all $x$ except arbitrarily small set of intervals. To avoid this, one would have to limit function change on given interval, which is problematic to express in conventional notation.
