Here's something which is likely mathematically 'trivial' in an analytic sense, but which is leading to some difficulties for me as part of a much larger discrete problem so I thought I'd see if someone could help in this aspect. The problem is to evaluate the usefully correct value of these equations (coordinate mappings) and each of their first partial derivatives at the singular values. The equations are of real, scalar valued continuous variables over the following domains: -1 <= x,e,z,r,s <= +1 though x,e,z are NOT linearly independent and they range over a smaller space than an [-1:+1] axis cube. The mapping equations are: r=( -2*(1+x)/(e+z))-1 s=(+2*(1+e)/(1-z))-1 There are singularities at every locus where (e+z) == 0, and only as a subset of those when z==+1 and e=-1. In the discrete cases of interest in the x;e;z sub domain, the singularities occur only where: e == -z, and simultaneously x == -1. In all cases of interest for the above mapping equations, I believe that the problem is a real-valued interpretation of a limit as the numerator and denominator both approach 0, so the result is 0/0 at the singularity. The equations yield well defined differentiable values over the deleted (real) neighborhood of each singularity. So were these complex domain equations, the singularities should be removable. The partials are as follows, and run into similar problems: dr/dx=-2/(z+e) dr/de=2*(x+1)/(z+e)^2 dr/dz=2*(x+1)/(z+e)^2 ds/de=2/(1-z) ds/dz=2*(e+1)/(1-z)^2 So, the question/problem is: Where e=-z, and x=-1, define: r(x,e,z) s(e,z) dr/dx, dr/de dr/dz ds/de ds/dz Removing the singularity is well known wrt. integration, and with respect to complex functions, though the theory and practice of doing it with respect to discrete evaluation and partial differentiation of real valued functions of multiple non-orthogonal variables isn't as familiar to me. In this case, it seems like the values ought to be easily determined analytically as limits. Most of all, I'm trying to get it simplified so that when discretely calculated the result is found without using numerically unstable evaluations of the function with small coordinate deltas to evaluate the value of the function in the deleted neighborhood of the singularity. I believe I may be looking for either a constant limit or some kind of change-of-variable type of evaluation that lets me evaluate the value without numerically undesirable calculations involving small deltas or ratios of small numbers. I suppose it's possible to define a finite series to well interpolate the value of the function throughout the neighborhood of the singularity even though analytic continuation in general doesn't apply (AFAIK) in the real domain. Thanks in advance for any elucidations about this!