  In this paper, we consider non-local integro-differential equations under
certain natural assumptions on the kernel, and obtain persistence of H\"{o}lder
continuity for their solutions. In other words, we prove that a solution stays
in $C^\beta$ for all time if its initial data lies in $C^\beta$. This result
has an application for a fully non-linear problem, which is used in the field
of image processing. The proof is in the spirit of the paper [18] of Kiselev
and Nazarov where they established H\"{o}lder continuity of the critical
surface quasi-geostrophic (SQG) equation.
