  We study local filters for the Lipschitz property of real-valued functions
$f: V \to [0,r]$, where the Lipschitz property is defined with respect to an
arbitrary undirected graph $G=(V,E)$. We give nearly optimal local Lipschitz
filters both with respect to $\ell_1$-distance and $\ell_0$-distance. Previous
work only considered unbounded-range functions over $[n]^d$. Jha and
Raskhodnikova (SICOMP `13) gave an algorithm for such functions with lookup
complexity exponential in $d$, which Awasthi et al. (ACM Trans. Comput. Theory)
showed was necessary in this setting. We demonstrate that important
applications of local Lipschitz filters can be accomplished with filters for
functions with bounded-range. For functions $f: [n]^d\to [0,r]$, we circumvent
the lower bound and achieve running time $(d^r\log n)^{O(\log r)}$ for the
$\ell_1$-respecting filter and $d^{O(r)}\text{polylog } n$ for the
$\ell_0$-respecting filter. Our local filters provide a novel Lipschitz
extension that can be implemented locally. Furthermore, we show that our
algorithms have nearly optimal dependence on $r$ for the domain $\{0,1\}^d$. In
addition, our lower bound resolves an open question of Awasthi et al., removing
one of the conditions necessary for their lower bound for general range. We
prove our lower bound via a reduction from distribution-free Lipschitz testing
and a new technique for proving hardness for adaptive algorithms. We provide
two applications of our local filters to arbitrary real-valued functions. In
the first application, we use them in conjunction with the Laplace mechanism
for differential privacy and noisy binary search to provide mechanisms for
privately releasing outputs of black-box functions, even in the presence of
malicious clients. In the second application, we use our local filters to
obtain the first nontrivial tolerant tester for the Lipschitz property.
