Okay, the math topic on my brain right this second is the question of whether or not it is reasonable to say a curve has multiple tangent lines at a point. I'll come right out and say I think not. It seems to me that for a point P on a curve C to have a tangent line with slope m, it must be the case that for all epsilon>0, there exists a delta>0 such that for any point Q on the curve C, if Q is within delta of P then the slope of the secant line through P and Q is within epsilon of m. This definition makes it possible to only have one tangent line.

Really, that has to be the accepted definition. But I have to go check.

In particular, the question comes up when you consider nice curves (polynomial curves, for instance) with self-crossings. I think it only reasonable to say the curve has no tangent line at the self-crossing (unless, of course, the crossings are all in the same "direction"). However, the calculus text I'm teaching out of appears to have no problem with saying the curve has multiple tangent lines at self-crossings. This latter conception leads, I think, to numerous problems, and will cause difficulties with the basic notion of differentiability. Or maybe I'm mixing up the two. Or maybe it's all just a matter of convention. I need to go check some reliable sources.