Consider an ambient metric space $(\mathcal{X},\Vert\cdot\Vert_\infty)$. Let $\mathcal{B}_1 = \mathcal{B}_{\Vert\cdot\Vert_K}(0,1)\subseteq\mathcal{X}$ be the closed unit ball with respect to some norm $\Vert\cdot\Vert_K$. Denote the $\varepsilon$-covering number of $\mathcal{B}_1$ with respect to $\Vert\cdot\Vert_\infty$ by $\mathcal{N}(\varepsilon, \mathcal{B}_1,\Vert\cdot\Vert_\infty)$. That is, we can find a set of points $\{x_1,\dots,x_n\}\subseteq\mathcal{X}$ with $n = \mathcal{N}\left(\varepsilon, \mathcal{B}_1,\Vert\cdot\Vert_\infty\right)$ such that for all $x\in\mathcal{B}_1$, there exists $i\in[n]$ with \begin{equation} \Vert x - x_i\Vert_\infty\leq\varepsilon \end{equation} From the observation that the ball is closed and the covering number is defined with $\leq$ instead of $<$ (see equation above), I am tempted to assume that the covering number is a right-continuous function of $\varepsilon$. Is this assumption true?