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 \[ v\ \mapsto\ \int_\Omega \varphi(x,|Dv|)\,dx, \] where $\varphi$
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
$\varphi$ in terms of a single condition for the map $(x,t)\mapsto
\varphi(x,t)$, rather than separately in the $x$- and $t$-directions. Thus we
can obtain regularity results for functionals without assuming that the gap
between the upper and lower bounds is small, i.e. $\frac qp$ need not be close
to $1$. Moreover, for $\varphi(x,t)$ 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.