  We study the higher H\"older regularity of local weak solutions to a class of
nonlinear nonlocal elliptic equations with kernels that satisfy a mild
continuity assumption. An interesting feature of our main result is that the
obtained regularity is better than one might expect when considering
corresponding results for local elliptic equations in divergence form with
continuous coefficients. Therefore, in some sense our result can be considered
to be of purely nonlocal type, following the trend of various such purely
nonlocal phenomena observed in recent years. Our approach can be summarized as
follows. First, we use certain test functions that involve discrete fractional
derivatives in order to obtain higher H\"older regularity for homogeneous
equations driven by a locally translation invariant kernel, while the global
behaviour of the kernel is allowed to be more general. This enables us to
deduce the desired regularity in the general case by an approximation argument.
