# 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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### Domain, Co-Domain & Range of a Function

I'm a little confused between the difference between the range & co-domain of a function. Are they not the same thing (i.e. all possible outputs of the function)?
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### Is there a Cantor-Schroder-Bernstein statement about surjective maps?

Let $A,B$ be two sets. The Cantor-Schroder-Bernstein states that if there is an injection $f\colon A\to B$ and an injection $g\colon B\to A$, then there exists a bijection $h\colon A\to B$. I was ...
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### Intuition behind Cantor-Bernstein-Schröder

The book I am working from (Introduction to Set Theory, Hrbacek & Jech) gives a proof of this result, which I can follow as a chain of implications, but which does not make natural, intuitive ...
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### Construct a monotone function which has countably many discontinuities

I read in a textbook, which had seemed to have other dubious errors, that one may construct a monotone function with discontinuities at every point in a countable set $C \subset [a,b]$ by enumerating ...
### Show that if $g \circ f$ is injective, then so is $f$.
The Problem: Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions. (a) Show that if $g \circ f$ is injective, then so is $f$. (b) If $g \circ f$ is surjective, must $g$ be surjective? ...