We consider now the graph of the function f (t) on **f. We will refer to the plane of C and **f as the C-plane and to the plane of the graph as the f-plane. The graph, as a set, may have a finite number of components. We will denote the values of f (t) on different components by **f. Each point with abscissa t on the graph represents an intersection between C and **f. There are two types of such intersections, depending essentially on whether the curves cross at the point of intersection. An ordinary point will be any point of intersection A such that in every neighborhood of A in the C-plane, **f meets both the interior and the exterior of C. Any other point of intersection between C and **f will be called a tangent point. This terminology will also be applied to the corresponding points in the f-plane. We can now prove several lemmas.

In some neighborhood in the f-plane of any ordinary point of the graph, the function f is a single valued, continuous function.

We first show that the function is single valued in some neighborhood. With the vertex at **f in the C-plane we assume that **f is the parametric location on C of an ordinary intersection Q between C and **f. In the f-plane the coordinates of the corresponding point are **f. We know that in the C-plane both C and **f are analytic. In the C-plane we construct a set of rectangular Cartesian coordinates u, v with the origin at Q and such that both C and **f have finite slope at Q. Near Q, both curves can be represented by analytic functions of u. In a neighborhood of Q the difference between these functions is also a single valued, analytic function of u. Furthermore, one can find a neighborhood of Q in which the difference function is monotone, for since it is analytic it can have only a finite number of extrema in any interval. Now, to find **f, one needs the intersection of C and **f near Q. But **f is just the curve **f translated without rotation through a small arc, for **f is always obtained by rotating C through exactly 90 `. The arc is itself a segment of an analytic curve. Thus if e is sufficiently small, there can be only one intersection of C and **f near Q, for if there were more than one intersection for every e then the difference between C and **f near Q would not be a monotone function. Therefore, **f is single valued near Q. It is also seen that **f, since the change from **f to **f is accomplished by a continuous translation. Thus **f is also continuous at **f, and in a neighborhood of **f which does not contain a tangent point.