Let g be continuous on an interval A and let F be the set of

Let g be continuous on an interval A and let F be the set of points where g fails to be one-to-one; that is, F = {x belongs to A: f(x) = f(y) for some y notequalto x and y belongs to A}. Show F is either empty or uncountable.

Solution

Given that g is continuous on the interval A.

g is not one to one in F

i.e. g(x) = g(y) for x not equal to y where x and y are in F.

Since g is not one to one throughout the interval, g cannot be either increasing or decreasing.

Hence g will be increasing as well as decreasing in the interval.

Therefore there are chances for local maxima or minima for g(x) in the interval.

Hence this is possible if f(x) = even powers of x which means not one to one for more than infinite points.