Limit sets of typical continuous functions

Given a metrizable compact topological n-manifold X with boundary and a finite positive Borel measure μ on X, we prove that for the typical continuous function , it is true that for every point x in a full μ-measure subset of X the limit set ω(f,x) is a Cantor set of Hausdorff dimension zero, f maps ω(f,x) homeomorphically onto itself, each point of ω(f,x) has a dense orbit in ω(f,x) and f is non-sensitive at each point of ω(f,x); moreover, the function x→ω(f,x) is continuous μ-almost everywhere.