Let F be a family of functions containing the function f. For example, F might be the family of potential functions of chapter 1 (see 1.1), which depended on two parameters a and b, or it might be the family of distance-squared functions on a curve in ℝn, where there are n parameters given by the coordinates of a point u ∈ ℝn (see 2.13). Changing the parameters a bit will ‘perturb’ f into a nearby function. In fig. 6.1 there are the graphs of some perturbations of f(t) = t5 (the axes are not drawn, but the t-axis is in each case horizontal).

The label k at the point of the graph where t = t0 indicates that the function has a turning point which is an Ak singularity at t = t0, or equivalently that the tangent line there is horizontal and has (k + 1)-point contact with the curve. Notice that sometimes two turning points can be on the same level, i.e. the function has the same value there, and sometimes, as with two A2s, this is not possible.

A family of functions containing f is also called an unfolding of f : the family unfolds to reveal all these functions which are f's close relations. Certain unfoldings contain all functions close to f (in a precise sense).