Sentence:  The fact that there can not be any limit points of the set except in closed intervals follows from the argument <WSD>used</WSD> 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 . 
Meaning of the word marked with <WSD> tags:  put into service; make work or employ for a particular purpose or for its inherent or natural purpose