I've come across the statement that the following function, referred to as Coulomb cost in optimal transport, is lower semicontinuous: $c : \mathbb{R}^{Nd} \rightarrow \mathbb{R}, c(x_1,.., x_N) = \sum_{1 \leq i < j \leq N}\frac{1}{|x_i - x_j|}$ (for example, this paper: https://flore.unifi.it/bitstream/2158/1070973/1/m2an150051.pdf, Remark 3.14).
It is assumed to be $\infty$ when $x_j = x_j$ for two indices $i \neq j.$ The function is continuous when $x_i \neq x_j, \forall 1 \leq i < j \leq N,$ so I'm wondering about the pathological case where it is $\infty$. If we take $N=2$ for simplicity, for every sequence $(x_n, y_n) \rightarrow (x,x), \liminf_{n \rightarrow \infty} c(x_n, y_n) = \infty = c(x,x),$ because $c(x_n, y_n)$ diverges. My confusion: Why is this specifically stated as lower semicontinuous and not simply continuous? We would also get $\limsup_{n \rightarrow \infty} c(x_n,y_n) \leq \infty = c(x,x),$ (actually "$\limsup_{n \rightarrow \infty} c(x_n,y_n) = \infty = c(x,x)$"), so why not simply continuous?
