Number of elements is odd infinity

Question

If we have one continuous function F(x), and if we define f(x)=F(x) on domain from open interval (a, b), and if F(a)=F(b)
If function f(x) is monotonically increasing from point a to point M, and monotonically decreasing from point M to point b
Can we assume there is odd infinity number of numbers for which f(x) is defined?
Because for every x there is one z where x,z are from domain of f(x) except for M?
Every element from that range has it's pair except for M

Answer 1

If your open interval has a cardinality which is odd infinity then presumably a similar argument would suggest that a half-open interval would have a cardinality which is an even infinity.

It is possible to find a bijection between a half-open interval and an open interval, or between a half-open interval and a closed interval: see How to define a bijection between $(0,1)$ and $(0,1]$? and the questions linked form it.

The bijection shows that these two cardinalities are in fact the same, which is why infinities are not described as odd or even.

Answer 2

Infinity cannot be even or odd. "Odd infinity" is a nonsensical phrase.