Sentence:  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 <WSD>function</WSD> in some neighborhood on either side of the tangent point . 
Meaning of the word marked with <WSD> tags:  (mathematics) a mathematical relation such that each element of a given set (the domain of the function) is associated with an element of another set (the range of the function)