# Tagged Questions

Elementary questions about functions, notation, properties, and operations such as function composition.

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### Isomorphic embedding of $L^{p}(\Omega)$ into $L^{p}(\Omega \times \Omega)$?

Let $(\Omega,\mu)$ be a finite measure space such that $\mu(\Omega)=1$. Suppose $1\leq p \leq \infty$. Let $\psi \colon L^p(\Omega) \to L^p(\Omega \times \Omega)$ be the map which maps $f$ onto the ...
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### Let $(f(x))^2$ and $(f(x))^3$ are $C^{\infty}$. Prove or disprove that $f$ is $C^{\infty}$.

Suppose $f(x), -\infty < x < +\infty$, is a real valued function such that both $(f(x))^2$ and $(f(x))^3$ are $C^{\infty}$. Must $f$ be $C^{\infty}$? I had seen this exercise somewhere ...
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### Function such that zeros$=$order of the derivative

Does there exist a function $f\in C^n(\mathbb{R},\mathbb{R})$ for $n\ge2$ such that $f^{(n)}$ has exactly $n$ zeros, $f^{(n-1)}$ has exactly $n-1$ zeros and so on ? Where $f^{(n)}$ is the nth ...
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### If $B(x+y)-B(x)-B(y)\in\mathbb Z$ can we add an integer function to $B$ to make it additive?

Given a function $B:\mathbb R\to\mathbb R$ satisfying $B(x+y)-B(x)-B(y)\in\mathbb Z$ for all real numbers $x$ and $y$, is there a function $Z:\mathbb R\to\mathbb Z$ such that $B+Z$ is an additive ...
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### 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 ...
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### A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
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### Why is it important to have a discrepancy between image and codomain?

A function $f:\mathbb{R}\rightarrow \mathbb{R}$ given by $f(x)=x^2$ has $\mathbb{R}_{\geq0}$ as its image and $\mathbb{R}$ as its codomain. What's the need for this discrepancy? Why don't we just ...
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### What does orthogonality mean in function space?

The functions $x$ and $x^2 - {1\over2}$ are orthogonal with respect to their inner product on the interval [0, 1]. However, when you graph the two functions, they do not look orthogonal at all. So ...