# Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

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### How to define a bijection between $(0,1)$ and $(0,1]$?

How to define a bijection between $(0,1)$ and $(0,1]$? Or any other open and closed intervals? If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that'...
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### What functions can be made continuous by "mixing up their domain"?

Definition. A function $f:\Bbb R\to\Bbb R$ will be called potentially continuous if there is a bijection $\phi:\Bbb R\to\Bbb R$ such that $f\circ \phi$ is continuous. So one could say a potentially ...
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### Overview of basic results about images and preimages

Are there some good overviews of basic facts about images and inverse images of sets under functions?
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### Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$...

I've been stumped by this problem: Find three non-constant, pairwise unequal functions $f,g,h:\mathbb R\to \mathbb R$ such that $$f\circ g=h$$ $$g\circ h=f$$ $$h\circ f=g$$ or prove that no ...
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### How do I prove that a function is well defined?

How do you in general prove that a function is well-defined? $$f:X\to Y:x\mapsto f(x)$$ I learned that I need to prove that every point has exactly one image. Does that mean that I need to prove the ...
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### Proving that a function is odd

Assume that there exists a function $f:\mathbb{R}\to\mathbb{R}$ that is bijective and satisfies $$f(x) + f^{-1}(x)=x$$ for all $x$. Here $f^{-1}$ is the inverse function. Show that $f$ is odd. This ...
### Function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ that takes each value in $\mathbb{R}$ three times? If not, how could I prove that such a function does not exist?