  We obtain sharp bounds for the modulus of continuity of the uncentered
maximal function in terms of the modulus of continuity of the given function,
via integral formulas. Some of the results deduced from these formulas are the
following: The best constants for Lipschitz and H\"older functions on proper
subintervals of $\mathbb{R}$ are $\operatorname{Lip}_\alpha ( Mf) \le (1 +
\alpha)^{-1}\operatorname{Lip}_\alpha( f)$, $\alpha\in (0,1]$. On $\mathbb{R}$,
the best bound for Lipschitz functions is $ \operatorname{Lip} ( Mf) \le
(\sqrt2 -1)\operatorname{Lip}( f).$ In higher dimensions, we determine the
asymptotic behavior, as $d\to\infty$, of the norm of the maximal operator
associated to cross-polytopes, euclidean balls and cubes, that is, $\ell_p$
balls for $p = 1, 2, \infty$. We do this for arbitrary moduli of continuity. In
the specific case of Lipschitz and H\"older functions, the operator norm of the
maximal operator is uniformly bounded by $2^{-\alpha/q}$, where $q$ is the
conjugate exponent of $p=1,2$, and as $d\to\infty$ the norms approach this
bound.
  When $p=\infty$, best constants are the same as when $p = 1$.
