  Let $(f, g)$ be a pair of complex analytic functions on a singular analytic
space $X$. We give ``the correct'' definition of the relative polar curve of
$(f, g)$, and we give a very formal generalization of L\^e's attaching result,
which relates the relative polar curve to the relative cohomology of the Milnor
fiber modulo a hyperplane slice. We also give the technical arguments which
allow one to work with a derived category version of the discriminant and Cerf
diagram of a pair of functions. From this, we derive a number of
generalizations of results which are classically proved using the discriminant.
In particular, we give applications to families of isolated ``critical
points''.
