Continuity of $f(x)=\inf_{d\in D}\inf_{c\in F} d(c,x)$ Suppose $D$ is a collection of metrics on a set $X$. $\mathcal T$ is the (smallest) topology generated by metrics in $D$. $F\subseteq X$ is closed in $(X,\mathcal T)$. Is the function
$$f:(X,\mathcal T)\to \Bbb R$$
$$f(x)=\inf_{d\in D}\inf_{c\in F} d(c,x)$$
continuous?

No, not necessarily.  Let $X$ be the closed interval $[0,2]$ in $\Bbb R$, and let $D$ be the set of metrics $\{d_j\mid j\ge 1\}$ on $X$, where
$$d_j(x,y):=|x^j-y^j|, \qquad j=1, 2, 3, \dots.$$
Since the map $x\mapsto x^j$ is a homeomorphism from $[0,2]$ to $[0,2^j]$, each $d_j$ generates the usual topology on $[0,2]$, so $\cal T$ is also the usual topology on $[0,2]$.  Then, $F:=\{0\}$ is closed in $(X,\cal T)$ but
$$
f(x)=\inf_{j} d_j(0,x) = \left\{\begin{array}{ll} 0, & 0\le x<1\\
x, & 1\le x\le 2\end{array}\right.
$$
is not continuous as a function from $(X,\cal T)$ to $\Bbb R$.
