Consider the metric space $(\mathbb{R}^{\mathbb{N}},\rho)$ with metric $$\rho(x,y) = \sum_{k=1}^\infty 2^{-k}\frac{|x_k - y_k|}{1 + |x_k - y_k|}.$$
Is $x\mapsto \sup_{k\geq 1}x_k$ continuous with respect to this metric?
Clearly, $$\sup_{k\geq 1}x_k - \sup_{k\geq 1}y_k \leq \sup_{k\geq 1}|x_k - y_k|$$
Is it possible to bound this by $\rho(x,y)?$
$x \mapsto \sup_{k \geq 1} x_k$ is not continuous for the topology of coordinatewise convergence. For example, consider for each $n$ the sequence $x^{(n)}$ given by $$x_k^{(n)} = \begin{cases} 1, \quad k = n \\ 0, \quad k \neq n\end{cases}$$ Then $\rho(x^{(n)},0) = 2^{-(n+1)} \to 0$ as $n \to \infty$ but $\sup_{k \geq 1} x_k^{(n)} = 1$ which does not converge to $0$ as $n \to \infty$. Hence your function is not continuous.