Let \(f : [ 0 , 1 ] \to \mathbb { R }\) be continuous with \(f ( 0 ) = f ( 1 )\)
Show that for each \(n \in \mathbb { N }\) there exist \(x , y \in [ 0 , 1 ]\) satisfying \(| x - y | = 1 / n\) and \(f ( x ) = f ( y )\)
Show that if \(h \in ( 0 , 1 / 2 )\) , but not of the form \(1 / n\) for some \(n \in \mathbb { N }\) , then there does not necessarily exist \(x , y \in [ 0 , 1 ]\) satisfying \(| x - y | = h\) and \(f ( x ) = f ( y )\)