Let $X$ be a metric space with metric $d$. Show that $d:X \times X \longrightarrow \mathbb{R}$ is continuous.
46834
show that a distance function is continuous
6415
Distance function is continuous in topology induced by the metric
90628
The metric spaces are Tychonoff spaces
153730
Show that $f(x)=x^2$ is continuous at $a=2$ using the $\delta-\epsilon$ definition of continuity.
75479
Proving that this function is continuous on $G\times G$
32359
Inner product of two continuous maps is continuous
106256
Continuous mapping on a compact metric space is uniformly continuous
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
98772
Continuity of a function in $\mathbb{R}$
42266
Show that $d(x,y)=\min \{1,|x-y|\}$ is a metric on $\mathbb R$
69777
Show that, for every metric $d$, the metrics $d/(1+d)$ and $\min\{1,d\}$ are equivalent
2093
If $f$ is continuous then $G$ is homeomorphic to $X$.
23845
How to prove the continuity of the metric function?
127034
Let $\rho(x,y)= \frac{d(x,y)}{1+d(x,y)}$, show that $\rho$ is a metric equivalent to $d$
118207
Prove that the product of two continuous functions is continuous
7402
Showing a compact metric space has a countable dense subset
16060
Showing this metric space is complete