approximation of CDF

You may need to clarify your terminologies.  My understanding is that
function *approximation* is for the situation where you have x1,..., xn and
f(x1), ..., f(xn), without any error (a numerical analysis problem).
Function *estimation* is when f() is not known, but estimated from data (a
statistical problem).  Sounds like you need estimation, not approximation.
(There's duality between the two, but they are different problems.)

As Prof. Ripley suggested, most density estimators give you proper
densities, so integrating those will give you a proper CDF, in addition to
the monotonicity property that you mentioned (e.g., f(-Inf)=0, f(Inf)=1).

Andy