  Let C be a closed subset of a topological space X, and let f : C --> X. Let
us assume that f is continuous and f(x) lies in C for every x in the boundary
of C.
  How many times can one iterate f?
  This paper provides estimates on the number of iterations and examples of
their optimality. In particular we show how some topological properties of f,
C, X are related to the maximal number of iterations, both in the case of
functions and in the more general case of set-valued maps.
