  Reproducing kernel Hilbert spaces (RKHSs) are very important function spaces,
playing an important role in machine learning, statistics, numerical analysis
and pure mathematics. Since Lipschitz and H\"older continuity are important
regularity properties, with many applications in interpolation, approximation
and optimization problems, in this work we investigate these continuity notion
in RKHSs. We provide several sufficient conditions as well as an in depth
investigation of reproducing kernels inducing prescribed Lipschitz or H\"older
continuity. Apart from new results, we also collect related known results from
the literature, making the present work also a convenient reference on this
topic.
