We consider equidistant Riemann approximations of stochastic integrals $\int_0^T f(B^H_s)dB^H_s$ with respect to the fractional Brownian motion with $H>\frac12$, where $f$ is an arbitrary function of locally bounded variation, hence possibly possessing discontinuities. We prove that properly normalised approximation error converge in the $L^2$-topology to a functional of the local time, and we provide rate of convergence for this approximation. As such, our results complements some recent advances on the topic as well as provides new methods for simulation of local times.