I haven't been able to do much progress on the following question:
Show that for any $f , g \in \mathcal { B } ( X ; Y )$ and $x \in X$, there holds
$\omega _ { f } ( x ) \leq 2 \rho _ { \text { sup } } ( f , g ) + \omega _ { g } ( x )$
Deduce that $\mathcal { B C } ( X ; Y )$ is closed with respect to $\rho_{sup}$.
Definitions:
For $E \subseteq X$, $\Omega _ { f } ( E ) = \sup \left\{ d _ { Y } \left( f ( x ) , f \left( x ^ { \prime } \right) \right) | x , x ^ { \prime } \in E \right\}$
$\begin{array} { c } { \mathcal { B C } ( X ; Y ) = \mathcal { B } ( X ; Y ) \cap \mathcal { C } ( X ; Y )} \\ { \rho _ { \text { sup } } ( f , g ) = \sup _ { x \in X } d _ { Y } ( f ( x ) , g ( x ) ) } \end{array}$
$\mathcal{B}(X,Y)$ and $\mathcal{C}(X,Y)$ are respectively the set of bounded and continuous functions from $X$ to $Y$.
$\omega _ { f } ( x ) = \inf _ { r > 0 } \Omega _ { f } \left( N _ { r } ( x ) \right)$, where $N_r(x)$ is a neighborhood of radius $r$ around $x$
What I did:
I wasn't able to do much. It seems to me that the boundedness comes into play when using/coming up with the supremum as it guarantees it's not infinity. I also tried to arrive at the inequality by the means of the triangle inequality, but I haven't been successful.
$d(f(x_1),f(x_2)) \leq d(g(x_1),g(x_2)) +2\sup d(f(x),g(x))$. Take sup over $x_1,x_2 \in N_r(x)$ and let $r \to 0$. Note that (by monotonicity) $\omega_f(x)$ is also $\lim_{r \to 0} \Omega_f(N_r(x))$. For the second part use the fa ct that $f$ is continuous at $f$ iff $\omega_f(x)=0$. Thus, if $f_n$'s are continuous and $f_n \to f$ w.r.t. $\rho_{sup}$ then $\omega_f(x) \leq 2\rho_{sup} (f_n,f) +0 \to 0$ for all $x$ so $f$ is continuous.