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We first show that the function is single-valued in some neighborhood.

With the vertex at Af in the C-plane we assume that Af is the parametric location on C of an ordinary intersection Q between C and Af.

In the f-plane the coordinates of the corresponding point are Af.

We know that in the C-plane both C and Af 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 Af 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 Af, one needs the intersection of C and Af near Q.

But Af is just the curve Af translated without rotation through a small arc, for Af is always obtained by rotating C through exactly 90-degrees.

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 Af near Q, for if there were more than one intersection for every E then the difference between C and Af near Q would not be a monotone function.

Therefore, Af is single-valued near Q.

It is also seen that Af, since the change from Af to Af is accomplished by a continuous translation.

Thus Af is also continuous at Af, and in a neighborhood of Af which does not contain a tangent point.