# 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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### 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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### I need a better explanation of $(\epsilon,\delta)$-definition of limit

I am reading the $\epsilon$-$\delta$ definition of a limit here on Wikipedia. It says that $f(x)$ can be made as close as desired to $L$ by making the independent variable $x$ close enough, but ...
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### How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$? [closed]

Prove that $\arccos x + \arccos(-x) = \pi$ when $x \in [-1,1]$. How do I prove this? Where should I begin and what should I consider?
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### Is There a Natural Way to Extend Repeated Exponentiation Beyond Integers?

This question has been in my mind since high school. We can get multiplication of natural numbers by repeated addition; equivalently, if we define $f$ recursively by $f(1)=m$ and $f(n+1)=f(n)+m$, ...
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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 ...
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### how to solve binary form $ax^2+bxy+cy^2=m$, for integer and rational $(x,y)$
solve $3x^2+3xy-5y^2=55$ using number theory tools ,i have found the following $\Delta=3^2+4(5)(3)=9+60=69$ $d=69,u=1$ $w_{69}=\frac{1+\sqrt{69}}{2}$ \$O_{69}=\theta_{-11}=[1,\frac{1+\sqrt{69}...