You don't have access to the "real" continuous distribution, though, but only a finite sample of points from it. Most non-parametric ways of modeling that are going to require some choice of smoothing parameter, either something like a histogram bin width, or a kernel-regression bandwidth (in either case, you can use the data to choose one, using cross-validation).
How about working with the distribution function instead of densities? That way you still have the sampling, but you don't have to decide on a bin size.
