Aiming at clarifying recent advances on disturbance rejection, we analyze a class of non-differentiable disturbances. We expose some subtle but fundamental differences between the fractional order (FO) and integer order (IO) integral sliding modes (ISM) to control a general class of nonlinear dynamical systems. This comparative study suggests that our proposed FOISM outperforms the extremely popular IOISM scheme when the dynamical system is subject to Hölder continuous but not necessarily differentiable disturbances. The drawbacks and advantages of the FOISM are discussed, including its capacity to reject Hölder disturbances with chattering alleviation. It is shown that the proposed method provides a uniform continuous control signal assuring robustness and invariance for any initial condition. Simulation results reveal the structural differences of these schemes, showing the viability of our proposal.