The discussion revolves around the question of whether a one-to-one function defined from ℝ to ℝ is necessarily monotone. Participants are exploring the implications of this statement and examining various examples and counterexamples.
The discussion is active, with participants sharing examples and engaging in clarifying questions. Some guidance has been offered regarding the need for a complete definition of the function across its entire domain. Multiple interpretations of monotonicity are being explored, particularly in relation to the examples provided.
Participants note the importance of defining the function on the entirety of ℝ and consider the implications of discontinuities in the context of monotonicity. There is also mention of philosophical considerations regarding the visualization of functions.
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?andrewkirk said:You have not defined the value of f(x) outside of I, so the example does not meet the requirements of the question, which include that the domain be all of ##\mathbb{R}##.
I think the statement is false, but a little more work is needed to produce a counterexample.
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.)andrewkirk said:(albeit harder to visualize).
Of course you could have f(x) = x outside of the interval (-1, 1) and f(x) = -x on the interval (-1, 1) .HaLAA said: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?