We turn now to the set of tangent points on the graph. This set must consist of isolated points and closed intervals. The fact that there can not be any limit points of the set except in closed intervals follows from the argument used in Lemma 1, namely, that near any tangent point in the C-plane the curves C and **f are analytic, and therefore the difference between them must be a monotone function in some neighborhood on either side of the tangent point. This prevents the occurrence of an infinite sequence of isolated tangent points.

In some neighborhood of an isolated tangent point in the f-plane, say **f, the function **f is either double valued or has no values defined, except at the tangent point itself, where it is single valued.

A tangent point Q in the C-plane occurs when C and **f are tangent to one another. A continuous change in t through an amount e results in a translation along an analytic arc of the curve **f. There are three possibilities: (a) **f remains tangent to C as it is translated; (b) **f moves away from C and does not intersect it at all for **f; (c) **f cuts across C and there are two ordinary intersections for every t in **f. The first possibility results in a closed interval of tangent points in the f-plane, the end points of which fall into category (b) or (c). In the second category the function **f has no values defined in a neighborhood **f. In the third category the function is double valued in this interval. The same remarks apply to an interval on the other side of **f. Again, the analyticity of the two curves guarantee that such intervals exist. In the neighborhood of an end point of an interval of tangent points in the f-plane the function is two valued or no valued on one side, and is a single valued function consisting entirely of tangent points on the other side.