Approximation of integration of piecewise continuous function at discrete points

Question

I have an integral of the form \begin{equation} \int_{0}^{1} f(x)dx, \end{equation} where $f$ is a continuous function that is not differentiable at countable number of points, defined on $[0,1]$. Suppose I take points $t_k = \frac{k-1/2}{4}, k=1,2,3,4$. Is there a way to approximate the above integral at these points? If $f$ was instead twice differentiable, then I know it can be approximated as \begin{equation} \int_{0}^{1} f(x)dx \approx \frac{1}{4}\sum_{k=1}^{4} f(t_k). \end{equation} I am looking for similar result that will be true for continuous function which is not differentiable at countable number of points.