  Model theory, machine learning, and combinatorics each have generalizations
of VC-dimension for fuzzy and real-valued versions of set systems. These
different dimensions define a unique notion of a VC-class for both fuzzy sets
and real-valued functions. We study these VC-classes, obtaining generalizations
of certain combinatorial results from the discrete case. These include
appropriate generalizations of $\varepsilon$-nets, the fractional Helly
property and the $(p,q)$-theorem.
  We then apply these results to continuous logic. We prove that NIP for metric
structures is equivalent to an appropriate generalization of honest
definitions, which we use to study externally definable predicates and the
Shelah expansion. We then examine distal metric structures, providing several
equivalent characterizations, in terms of indiscernible sequences, distal
types, strong honest definitions, and distal cell decompositions.
