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Sorry, this is my 3rd time for asking for help in a week and a half. I just have completely lost any intuition I had for math with all the time constraits I have.
a.)Prove that a continuous function, f:M->R, all of whose values are integers, is constant provided that M is connected.
b.)What if all the values are irrational?
M is connected if it has no separation. A separation is defined as 2 nonempty open sets A and B such that A U B=M and A [tex]\cap[/tex]B= the empty set.
Ny interpretation for f is constant is that it has only one element in it's image. f(x)=c for all x.
I proved a.) using contradiction. I assumed M was connected and f was not constant. I let A=the set of all x in M such that f(x)=c and B=the set of all x in M such that f(x) didn't equal c. I proved that both A and be were open and came to the conclusion that there was a separation. I can't use the same logic for part b.) as proving A was open required that the distance between each integer be a discrete value.
I'm assuming the answer for b.) is no but I don't know how to prove it.
Homework Statement
a.)Prove that a continuous function, f:M->R, all of whose values are integers, is constant provided that M is connected.
b.)What if all the values are irrational?
Homework Equations
M is connected if it has no separation. A separation is defined as 2 nonempty open sets A and B such that A U B=M and A [tex]\cap[/tex]B= the empty set.
Ny interpretation for f is constant is that it has only one element in it's image. f(x)=c for all x.
The Attempt at a Solution
I proved a.) using contradiction. I assumed M was connected and f was not constant. I let A=the set of all x in M such that f(x)=c and B=the set of all x in M such that f(x) didn't equal c. I proved that both A and be were open and came to the conclusion that there was a separation. I can't use the same logic for part b.) as proving A was open required that the distance between each integer be a discrete value.
I'm assuming the answer for b.) is no but I don't know how to prove it.