With the above results we can make the following remarks about the graph of f. First, for any value of t for which all values of f (t) are ordinary points the number of values of f (t) must be odd. For it is clear that the total number of ordinary intersections of C and **f must be even (otherwise, starting in the interior of C, **f could not finally return to the interior), and the center of rotation at t is the argument of the function, not a value. Therefore, for any value of t the number of values of f (t) is equal to the (finite) number of tangent points corresponding to the argument t plus an odd number.

The number of ordinary values of the function f (t) at t will be called its multiplicity at t.

The graph of f has at least one component whose support is the entire interval [0, T].

We suppose not. Then every component of the graph of f must be defined over a bounded sub-interval. Suppose **f is defined in the sub-interval **f. Now **f and **f must both be tangent points on the nth component in the f-plane; otherwise by Lemma 1 the component would extend beyond these points. Further, we see by Lemma 2 that the multiplicity of f can only change at a tangent point, and at such a point can only change by an even integer. Thus the multiplicity of **f for a given t must be an even number. This is true of all components which have such a bounded support. But this is a contradiction, for we know that the multiplicity of f (t) is odd for every t.