Intereting Posts

Topology on $R((t))$, why is it always the same?
Geometric meaning of the determinant of a matrix
Integral of rational function with higher degree in numerator
How many $N$ digits binary numbers can be formed where $0$ is not repeated
When can an infinite abelian group be embedded in the multiplicative group of a field?
Example of a non measurable function!
Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+1/3 }{x^3-x+1})^2 \mathrm{d}x$
Simplex algorithm : Which variable will go out of the basis?
Applications of Belyi's theorem
Series for envelope of triangle area bisectors
Asymptotic integral expansion of $\int_0^{\infty} t^{3/4}e^{-x(t^2+2t^4)}dt$ for $x \to \infty$
Graphing and differentiation
product of harmonic functions
Mollifiers: Approximation
About Mellin transform and harmonic series

Suppose $(M,d)$ is metric. I have proven that if $\psi\colon[0,\infty)\to[0,\infty)$ is non-decreasing, subadditive and satisfies $\psi(x)=0\iff x=0$ for $x\ge0$, then $$\rho(x,y)=\psi(d(x,y))$$ is a metric on $M$.

But I want to `tweak’ the restriction of this statement, I want to use differentiability of $\psi$. I found the following:

- If a subsequence of a Cauchy sequence converges, then the whole sequence converges.
- General Triangle Inequality, distance from a point to a set
- Prove that there exists a Cauchy sequence, compact metric space, topology of pointwise convergence
- Prove the map has a fixed point
- Translation invariant metric
- Basic questions about $\mathbb{Z}^{\mathbb{N}}$ with the product topology

Suppose $(M,d)$ is metric and $\psi\colon[0,\infty)\to[0,\infty)$ is

differentiable with continuous non-increasing derivative $\psi'$ and

$\psi(0)=0$. Then $\psi$ is non-decreasing and subadditive.

I have also proven this statement, but is $\rho=\psi(d(x,y))$ a metric in this situation? And if not, what is a sufficient condition on the derivative $\psi'$ to turn $\rho$ into a metric on $M$?

- Equivalent metrics determine the same topology
- Why are every structures I study based on Real number?
- Sequential characterization of closedness of the set
- Continuous images of open sets are Borel?
- If $id:(X,d_1)\to (X,d_2)$ is continuous then what will be $X$?
- Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
- Does $|x|^p$ with $0<p<1$ satisfy the triangle inequality on $\mathbb{R}$?
- A topology on the set of lines?

You need to explicitly exclude $\psi'(0)=0$ (and thus due to the range of $\psi$ it follows $\psi'(0)\gt 0$) or alternatively require $\psi'$ strictly decreasing in any interval $(0,\epsilon)$. Otherwise, $\psi$ might satisfy the starting conditions of your second statement, but can violate $\psi(x)=0\iff x=0$, because if $\psi'$ can be zero on a positive interval starting at $0$, $\psi$ would be constant on that interval.

Then, from the starting conditions, $\psi' > 0$ on an interval near $0$ and thus $\psi$ strictly increasing, so $\neq 0$ on that interval. Now, from the given range of $\psi$ and $\psi'$ non-increasing it follows that $\psi' \ge 0$ everywhere – if it was not, then you can derive that $\psi$ must eventually become negative, which is a contradiction.

That in turn implies $\psi$ is strictly increasing near $0$ and (not necessarily strictly) increasing everywhere, and thus the remaining needed precondition of the first statement $\psi(x)=0\iff x=0$ follows.

- Software for drawing and analyzing a graph?
- Please help: My MATLAB code for solving a 2D Schrödinger equation keep giving me weird output.
- Prove that $\int_0^\infty \frac{\ln x}{x^n-1}\,dx = \Bigl(\frac{\pi}{n\sin(\frac{\pi}{n})}\Bigr)^2$
- Infinity times $i$
- Cubic with repeated roots has a linear factor
- Nilpotent matrix and basis for $F^n$
- What's the definition of a “local property”?
- Contradiction! Any Symbol for?
- Normality of subgroups and Sylow groups under homomorphisms…
- Subrings of polynomial rings over the complex plane
- Convergence of a compound sequence
- Let $\alpha:\langle a,b\rangle \longrightarrow \mathbb{R}^2$ be a continuous function and injective
- Commutativity of iterated limits
- The kernel of the transpose of the differentiation operator – Solution check
- Probability/Combinatorics Problem. A closet containing n pairs of shoes.