Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
2,753
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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?
58
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Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist
I've been trying to solve the following problem:
Suppose that $f$ and $f'$ are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to\...
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Calculating the total number of surjective functions
It is quite easy to calculate the total number of functions from a set $X$ with $m$ elements to a set $Y$ with $n$ elements ($n^{m}$), and also the total number of injective functions ($n^{\underline{...
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No continuous function switches $\mathbb{Q}$ and the irrationals
Is there a way to prove the following result using connectedness?
Result:
Let $J=\mathbb{R} \setminus \mathbb{Q}$ denote the set of irrational numbers. There is no continuous map $f: \mathbb{R} \...
- 5,698
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3
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Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large \frac{1}{z}$ by definition discontinuous at $0$?
Is function $f:\mathbb C-\{0\}\rightarrow\mathbb C$ prescribed by $z\rightarrow \large{\frac{1}{z}}$ by definition discontinuous at $0$?
Personally I would say: "no". In my view a function can only ...
- 152k
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Surjectivity of $f:S\to S$ implies injectivity for finite $S$, and conversely
Let $S$ be a finite set. Let $f$ be a surjective function from $S$ to $S$.
How do I prove that it is injective?
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Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
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Do we really need polynomials (In contrast to polynomial functions)?
In the following I'm going to call
a polynomial expression an element of a suitable algebraic structure (for example a ring, since it has an addition and a multiplication)
that has the form $a_{n}x^{...
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There exists an injection from $X$ to $Y$ if and only if there exists a surjection from $Y$ to $X$.
Theorem. Let $X$ and $Y$ be sets with $X$ nonempty. Then (P) there exists an injection $f:X\rightarrow Y$ if and only if (Q) there exists a surjection $g:Y\rightarrow X$.
For the P $\implies$ Q part, ...
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Inverse of $f(x)=\sin(x)+x$
What is the inverse of
$$f(x)=\sin(x)+x.$$
I thought about it for a while but I couldn't figure it out and I couldn't find the answer on the internet.
What about
$$f(x)=\sin(a \cdot x)+x$$
where ...
- 119
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When functions commute under composition
Today I was thinking about composition of functions. It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.
But then ...
- 5,608
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On the functional square root of $x^2+1$
There are some math quizzes like:
find a function $\phi:\mathbb{R}\rightarrow\mathbb{R}$
such that $\phi(\phi(x)) = f(x) \equiv x^2 + 1.$
If such $\phi$ exists (it does in this example), $\phi$ can ...
- 141k
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Is there a bijective map from $(0,1)$ to $\mathbb{R}$?
I couldn't find a bijective map from $(0,1)$ to $\mathbb{R}$. Is there any example?
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Is there a way to get trig functions without a calculator?
In school, we just started learning about trigonometry, and I was wondering: is there a way to find the sine, cosine, tangent, cosecant, secant, and cotangent of a single angle without using a ...
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