# Questions tagged [functions]

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

6,290 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
1k views

### Multiplicative Identity analog for absolute value

Is there a standard name for the function: $$f(x) = \begin{cases} x & \text{if |x|≤1;}\\ 1/x & \text{if |x|>1;}\\ \end{cases}$$ And is there a potential application of this ...
• 211
408 views

### On injectivity of $F_N$

I am interested in the following problem: Let $F_N:\mathbb{N}\to\mathbb{N}$ be the function that maps a natural number $a$ to the least natural number more than or equal to $a$ for which there exists ...
• 3,598
394 views

### Approximating the area below average of a concave function

Given a non-decreasing concave function $f:[0,1]\rightarrow \mathbb{R}^+$. Define \begin{align*} F(n)=\displaystyle\sum_{i=1}^n \min\left\{\frac{1}{n},\frac{f\left(\frac{i}{n+1}\right)}{n+1}\right\} ...
• 257
746 views

### Compositional "square roots"

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
• 1,610
361 views

248 views

### When does the "Zetor function" converge?

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}.$$...
• 7,246
247 views

• 348
416 views

### Need your thoughts on a faster approximation for the signum function I came up with.

I recently was kind of fascinated with the whole signum function out of the blue, and was generally really curious of the already existing approximations of it. I really don't want to lag this down ...
• 177
121 views

### Is $\pi: \mathcal{C}^\infty (M,N) \to \mathcal{C}^\infty (S,N)$, $\pi(f) = \left. f\right|_{S}$ a quocient map in the $\mathcal C^1$ topology?

Let $M, N$ be smooth connected manifolds (without boundary), where $M$ is a compact manifold, so we can put a topology in the space $\mathcal C^\infty(M, N)$ using $\mathcal{C}^1$ Whitney Topology. ...
• 3,284
121 views

136 views

### Explain approximate lines in graph of this function

Sorry that this is a long question; the crux of it is that I want to know why lines appear in the graph of the function ($\varphi^\infty(x)$) I've defined. Define $\varphi(x)$ as follows: If the ...
• 3,343
367 views

### Maximum Elo Rating.

I'm trying to implement a variant of the Elo system, for a game I'm working on. Giving two players $A$ and $B$ with ratings $R_A$ and $R_B$ respectively, the expectation of $A; E_A$ is given by the ...
• 1,217
507 views

### A method for evaluating sums/discrete functions by assuming they can be made continuous and differentiable?

Suppose I had a function that satisfied the property $f(x)=f(x-1)+g(x)$. For any $x\in\mathbb N$, it is easy enough to see that this boils down to the statement $$f(x)=f(0)+\sum_{k=1}^xg(k)$$ If we ...
594 views

208 views

### Does the existence of an associative, injective function imply the underlying set has only one element?

Assume we have a function $f: \Sigma \times \Sigma \to \Sigma$ sucht that $f$ is associative and injective. Does this imply $|\Sigma| = 1$? My reasoning is the following: Let $x, y, z \in \Sigma$. By ...
• 117
265 views

### Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(f(x+y)) = f(x)+f(y)$

The problem is to find the set of all functions $f: \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ we have $f(f(x+y)) = f(x)+f(y)$. We first notice that $f(x)=x+c$ is a solution. ...
• 2,510
100 views

### Fun functional equation in a group: $af(ab)b=bf(ba)a$

Suppose we have a finite group $G$ and a function $f:G\to G$ that satisfies the following property for all $a,b\in G$: $$af(ab)b=bf(ba)a$$ Under what conditions (or, for what groups $G$) can such a ...
165 views

### Definitions which were chosen because we write functions on the left

Writing function application as a left action has been baked into mathematics since the Bernoullis in the 1700s (cf. this MSE question). Because of this tradition, a lot of mathematics notation has ...
107 views

### Let function $g:A\to A$, $\:\:\:g^{2016}=g\circ g\circ g...\circ g=id_A$ , Prove that $g$ is Inverse function

Let function $g:A\to A$, $\:\:\:g^{2016}=g\circ g\circ g...\circ g=id_A$ , Prove that $g$ is Inverse function My attempted : $g$ is Inverse function if it's Bijection to prove $g$ is one-to-one ...
• 73
4k views

### What's the function poly() ??

In cryptographic context, we often observe the function $\mathsf{poly}$. For example , let $n$ be an integer, this function is called in a manner such as $\mathsf{poly}$(n). What's the exact ...
• 829
1k views

### Inverse of a product of real functions

Given $F(x) = L(x)G(x)$, with $L$ and $G$ real function strictly greater than zero. Suppose that F and G are decreasing functions (so that $F^{-1}$ and $G^{-1}$ exists). What can we say about the ...
• 131
601 views

• 7,780
3k views

### Log-concave functions whose sums are still log-concave: possible to find a subset?

Rationale: I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest. It is well-known that sums of log-...
• 61
280 views

### expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right)$$ I am wondering if one can expand ...
• 552
2k views

### Closed form for sine graphic rotated by 45 degrees?

Is there a non-parametric closed form for a function looking like a sine rotated 45 degrees? I have encountered also a similar question but it asks for a function resembling the rotated sine, but not ...
• 9,302
116 views

### Function for which it is unknown whether it is continuous

Is there any function $f:\mathbb R\rightarrow \mathbb R$ for which at least some values are known but it is unknown whether $f$ is continuous or not? Edit: I am looking for examples from actual ...
### Prove that if $2^x,3^x, 5^x, 7^x, 11^x ...$ are all integers then $x$ is an integer as well
How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ...$ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...