The nonparametric regression with a random design model is considered. We
want to recover the regression function at a point x where the design density
is vanishing or exploding. Depending on assumptions on the regression function
local regularity and on the design local behaviour, we find several minimax
rates. These rates lie in a wide range, from slow l(n) rates where l(.) is
slowly varying (for instance (log n)^(-1)) to fast n^(-1/2) * l(n) rates. If
the continuity modulus of the regression function at x can be bounded from
above by a s-regularly varying function, and if the design density is
b-regularly varying, we prove that the minimax convergence rate at x is
n^(-s/(1+2s+b)) * l(n)
