let f ab be continuous on ab where Z is the integers Show t

let f: (a,b) ---> be continuous on (a,b) where Z is the integers. Show that f must be a constant function

Solution

1st method:(thoeriticaly)

If the domain X is connected and f is continuous, then f(X) is also connected. The only connected subsets of the integers are sets containing at most one point. Hence f(X) (which is presumably non-empty) has exactly one point and f is constant on X.

so f is a constant function.

OR

2nd method:

let us assume that

y₁=f(x₁) and

y₂=f(x₂)

and y₁ <= y₂   without loss of generality

Then by the intermediate value theorem, f must take all the values in [y₁,y₂].

however, range is integer this can only be true when f takes only one value at a time.

so f is constant.