For the proof of Perron–Frobenius theorem, the book "Non-negative Matrices and Markov Chains", defines the function $r\colon\mathbb{R}^n\to \mathbb{R}$ as $$r(\boldsymbol x) := \min_{j \in \{ 1, \ldots, n \}} \frac{\sum_{i} x_i t_{ij}}{x_j},$$ where $t_{ij}$'s are components of a non-negative matrix $T \in \mathbb R^{n \times n}$. It claims $r(\boldsymbol x)$ is an upper semi-continuous function but I cannot see why.
Given the definition, we have to show that $\limsup_{k\to \infty} r(\boldsymbol x_k)\leq r(\boldsymbol x_0)$ for any sequence of $\boldsymbol x_k\to \boldsymbol x_0$. But I have no idea how to prove that. I suspect this is only true on the compact subspace of unit vectors but not sure.