# Tag Info

### Why write codomain instead of image when defining a function.

Here is an easy example. If $A$ is a set, then $\mathcal P(A)$ is the power set of $A$, i.e. $\{B\mid B\subseteq A\}$. In many contexts, we want to identify $\mathcal P(A)$ with $2^A$, that is the set ...

### How to prove that $f:B\to \mathbb{N}$ by $f(2^k\cdot 3^n)=nk$ with $n,k\in\mathbb{N}$ is a function.

Uniqueness of prime factorization...

### There exists a function that satisfies $\sum_{n=1}^\infty |f^{[n]}(x) - f^{[n]}(y)| < \infty$ but is not a contraction?

let $$f(x) =\begin{cases} 2x + 1 & 0 \le x < 1 \\ 0 & \text{ otherwise} \end{cases}$$ That's not a contraction (consider $x = 0, 0.5$ being sent to $1, 2$), but $f^{}$ is everywhere $0$...

### An Function that turns $\mathbb Z$ into $\mathbb Q$

Suppose $f:\Bbb Z\to\Bbb Q$ is monotonically increasing. If $f$ is constant, then its range is certainly not the whole of $\Bbb Q$. So suppose $f$ is not constant. Then there exists $n\in\Bbb Z$ such ...

1 vote
Accepted

### A piecewise function is continuous from a locally compact Hausdorff to metric space.

Clearly $\omega'$ is continuous on $\{f+g > 0\}$. If $x\in \overline{\{f+g > 0\}}$, take a net $x_\alpha \in \{f+g > 0\}$ such that $x_\alpha\to x$. Then from the inequality \$|\omega'(x_\...
1 vote

I found a definition with floor function that makes the function continuous on entire real line, but this definition is only disguise of a piecewise definition. $$f(x)=\frac{\arctan(2 \tan (x))+\pi \... 1 vote ### How to define this function so that it is continuous? Maybe better to leave it in its original implicit form, so that the range is continuous. Converted to its implicit function$$ \tan (x~y)= 2~\tan(x)$$and directly plotted on ... 1 vote ### How to define this function so that it is continuous? The question is about the simplest value for the multi-valued function$$ y = f(x) := \text{arctan}(2\tan(x))/x. \tag1 $$A natural place to start is to define the two-variable function$$ g(x, y) = \...

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