Let $f$ be a concave function and D a metric of $M$. Prove that $f(D)$ is also a metric of $M$.
72033
Is this proof sufficient to show that a concave function of a metric is also a metric?
49721
Prove the empty set is closed for a metric space (X,d).
123061
$X$ metric separable then $C(X)$ separable
40438
Proving that closed (and open) balls are convex
83163
If $X$ is a connected metric space, then a locally constant function $f: X \to $ M, $M $ a metric space, is constant
121359
Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric.
144001
If $f$ is continuous and $\,f\big(\frac{1}2(x+y)\big) \le \frac{1}{2}\big(\,f(x)+f(y)\big)$, then $f$ is convex
9451
Show bounded and convex function on $\mathbb R$ is constant
114047
Metric $d(x,y)=\frac{|x-y|}{\sqrt{1+x^2}\sqrt{1+y^2}}$ on $\mathbb{R}$
117716
Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
69777
Show that, for every metric $d$, the metrics $d/(1+d)$ and $\min\{1,d\}$ are equivalent
71619
Proving that a convex function is Lipschitz
64172
Show that if $f$ is analytic in a domain $D$ and $|f(z)|$ is constant in $D$, then the function $f(z)$ is constant in $D$
73469
Continuity of the function $x\mapsto d(x,A)$ on a metric space
102959
Show that $f$ is convex if and only if $f\left( \sum_{i=1}^m\lambda_ix_i \right) \leq \sum_{i=1}^m\lambda_if(x_i)$
45370
Is it true that $|d(x, z) - d(y, z)| \leq d(x, y)$ and how do I prove it?
140383
Concave implies subadditive