  We study the regularity of solutions of functional equations of a generalized
mean value type. In this paper we give sufficient conditions for the regularity
by using hypoellipticity which is a concept of the theory of partial
differential equations. Further we also give an affirmative answer to a
conjecture of H.\'Swiatak. A part of the results was announced in the
comprehensive paper [8] on series of our joint works. To prove the regularity
of solutions of functional equation is in general one of central problem in the
theory of functional equations.

Functional Analysis;Functional Analysis