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## Homework Statement

A one to one function f: ℝ→ℝ is monotone, True or False

## Homework Equations

## The Attempt at a Solution

I think the statement is false, for example: Let I =[0,1]∪[2,3] f(x)=x if x∈[0,1], f(x)=5-x,x∈[2,3]

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- Thread starter HaLAA
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- #1

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A one to one function f: ℝ→ℝ is monotone, True or False

I think the statement is false, for example: Let I =[0,1]∪[2,3] f(x)=x if x∈[0,1], f(x)=5-x,x∈[2,3]

- #2

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I think the statement is false, but a little more work is needed to produce a counterexample.

- #3

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Let f(x)=x, x is rational, f(x)=-x,x is irrational, the function is one to one,but it is jumping. Does this example apply?

I think the statement is false, but a little more work is needed to produce a counterexample.

- #4

fresh_42

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- #5

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By the way, it's possible to extend your function in the OP to a non-monotone, injective function that has the entirety of ##\mathbb{R}## as domain. But the rational/irrational one in post 2 is easier to specify (albeit harder to visualize).

- #6

fresh_42

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Since both are dense, it's just a big X. And it leads to interesting philosophical questions: what one draws is always discrete for you put carbon atoms on the paper. (I apologize, if that remark should be regarded as improper.)(albeit harder to visualize).

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WWGD

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- #8

SammyS

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Of course you could have f(x) = x outside of the interval (-1, 1) and f(x) = -x on the interval (-1, 1) .Let f(x)=x, x is rational, f(x)=-x,x is irrational, the function is one to one,but it is jumping. Does this example apply?

or even simpler :

f(1) = -1, f(-1) = 1, otherwise, f(x) = x.

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