Fit_continuum fitting method is not robust to noise

Using fitting.fit_continuum to fit a simple linear continuum to two continuum windows gives significantly different slopes for two different realizations of the noise. The chi^2 (by-eye) value sometimes appears to be unacceptable. Here are two examples for different realizations of the noise, in the following continuum windows:
fitwindows1=[(5125.*u.AA, 5175.*u.AA),(5225.*u.AA, 5275.*u.AA)]


Start by reproducing the issue with fitting.fit_continuum and the supplied fitwindows1 values, using multiple noise realizations. Compare the resulting slopes and by-eye chi^2 values; done means the fitting behavior is acceptably consistent across those realizations.

Written by the indexing model from the issue text.