Prove that a continuous function f M rightarrow R all of who

Prove that a continuous function f: M rightarrow R, all of whose values are integers, is constant provided that M is connected. What if all the values are irrational?

Solution

If f: M->R is continuous and M is connected then f(M) is connected in R. Hence f(M) is an interval because it is a connected subset of R. Since M has only a finite number of points, the only interval it can be mapped to is a single point, hence f is a constant function.

B) We know that the irrational numbers are uncountable.

aslo f(M) is an interval,F(M) contains rational as well as irrational number.

So we can defined a identity map .which is also contineus.

that means f may or may not constant map.