It's known (and not hard to prove, using C^\infty bump functions --- I think this may have been originally due to Hassler Whitney) that every closed set in $R^n$ is the zero set of a C^\infty function. So, you can have two C^\infty surfaces intesecting in R^3 even intersect in much wilder sets than a simple arc --- e.g. they can intersect in a set that is connected but not locally connected, or whatever. If you're concerned about how the two surfaces cross, the construction allows you to independently choose the sign of the function in any component of the complement. On Dec 8, 2010, at 3:05 PM, Bill Gosper wrote:

Good grief, tutor Julian has plotted two infinitely differentiable surfaces intersecting in a simple arc that is not differentiable. --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun