Cardinality of the set $X$ 
Let $X$ be a set with the property that for any two metrics $d_1$ and $d_2$ on $X$, the identity map $\operatorname{id} : (X, d_1) \rightarrow (X,d_2)$ is continuous. Which of the followings are true?

(a) $X$ must be a singleton.
(b) $X$ may be any finite set.
(c) $X$ cannot be infinite.
(d) $X$ may be infinite but not uncountable.
I have tried by assuming $d_2$ is a discrete space and $d_1$ is not discrete. Suppose $X$ has more than one element. Since every singleton set is open in $(X,d_2)$ but not in $(X,d_1)$, this contradicts the definition of continuity.
Please check my solution and if you found any mistakes then correct me.
 A: This is not a complete answer, but at a partial solution.
(b) is in fact correct, i.e., $X$ can be any finite set, for singletons this is obvious. Let $X$ be a finite set with at least two elements. If $d$ is a metric on $X$, then we claim that $(X,d)$ is discrete. 
Let $d$ be a metric on a finite set $X$ with $cardX\geq 2$, then for any $x \in X$, the set $M=\{d(x,y)\ |\ y \not=x\}$ is a finite set with at least one element. Hence, it has a minimum, say $m$. Then $\{x\}=d(x,\frac{m}{2})$ and thus $(X,d)$ is discrete.
(d) To see that the assumption for an infinite set is not always true, consider $\Bbb{Q}$ with the standard metric $\rho$ and the discrete metric $d$. Then, clearly $\rho$ is not discrete, so that $id_\Bbb{Q}:(\Bbb{Q},\rho)\rightarrow (\Bbb{Q},d)$ is not continuous. 
If $X$ is any infinite set, then there is an injection $f$ from $\Bbb{Q}$ to $X$ and one can define two inequivalent metrics on $f[\Bbb{Q}]$ by translating the metrics $d$ and $\rho$ from $f$.
In other words if $X$ is any infinite set, and these two extended metrics (call them $\rho_f$ and $d_f$) on $f[\Bbb{Q}]$ can be extended to two metrics on $X$, then $X$ won't satisfy the assumption, because restriction of both the domain and codomain of a continuous function leaves one with a continuous function. In particular if $X$ is countable, then we may assume that $f$ is a bijection so that metrics $\rho_f$ and $d_f$ are not equivalent, or equivalently $id:(X,\rho_f)\rightarrow (X,d_f)$ is not continuous.
At the moment I can't seem to come up with an example of an infinite set $X$ where all metrics are discrete and I'm not entirely sure whether such a space exists. Maybe someone here knows a nice example.
