On monotonic functions from the unit interval into Banach spaces

We show that if there exists an increasing function f from [0,1] into a Banach lattice X such that for some ε>0 the set D(f,ε) of all points where the oscillation of f is at least ε is infinite, then X contains a subspace isomorphic to the space C(D(f,ε)) of all real continuous functions on D(f,ε).