Sentence:  The fact that there can not be any limit points of the set except in <WSD>closed</WSD> 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 . 
Meaning of the word marked with <WSD> tags:  an interval that includes its endpoints