ACTA issues

Continuity implies differentiability for solutions of functional equations --- even with few variables

Antal Járai

Acta Sci. Math. (Szeged) 67:3-4(2001), 719-734

Abstract. It is proved that --- under certain conditions --- continuous solutions $f$ of the functional equation $$ f(x)=h(x,y,f(g_1(x,y)),\ldots,f(g_n(x,y))), (x,y)\in D\subset{{\msbm R}^s}\times{\msbm R}^l, $$ are ${{\cal C}^\infty }$, even if $1\le l\le s$. As a tool we introduce new function classes which --- roughly speeking --- interpolate between differentiable and continuous functions.

AMS Subject Classification (1991): 39B05, 26B05

Received October 27, 2000. (Registered under 2813/2009.)