Pyramidal vectors and smooth functions on Banach spaces

We prove that if $X$, $Y$ are Banach spaces such that $Y$ has nontrivial cotype and $X$ has trivial cotype, then smooth functions from $X$ into $Y$ have a kind of ``harmonic" behaviour. More precisely, we show that if $\,\Omega$ is a bounded open subset of $X$ and $f:{\overline{\Omega }}\to Y$ is $C^{1}$-$\,$smooth with uniformly continuous Fréchet derivative, then $f(\partial \Omega )$ is dense in $f({\overline{\Omega }})$. We also give a short proof of a recent result of P. Hájek.