  We establish local $C^{1,\alpha}$-regularity for some $\alpha\in(0,1)$ and
$C^{\alpha}$-regularity for any $\alpha\in(0,1)$ of local minimizers of the
functional \[ v\ \mapsto\ \int_\Omega \phi(x,|Dv|)\,dx, \] where $\phi$
satisfies a $(p,q)$-growth condition. Establishing such a regularity theory
with sharp, general conditions has been an open problem since the 1980s. In
contrast to previous results, we formulate the continuity requirement on $\phi$
in terms of a single condition for the map $(x,t)\mapsto \phi(x,t)$, rather
than separately in the $x$- and $t$-directions. Thus we can obtain regularity
results for functionals without assuming that the gap $\frac qp$ between the
upper and lower growth bounds is close to $1$. Moreover, for $\phi(x,t)$ with
particular structure, including $p$-, Orlicz-, $p(x)$- and double phase-growth,
our single condition implies known, essentially optimal, regularity conditions.
Hence, we handle regularity theory for the above functional in a universal way.
