Show that $f(x_{0}\cdots x_{n}) = (\frac{x_{0}}{x_{i}}\cdots \frac{x_{i-1}}{x_{i}},\frac{x_{i+1}}{x_{i}},\cdots \frac{x_{n}}{x_{i}})$ is continuous.

Question

How to show that the function $f_{i}:D_{i}\subseteq \mathbb{R}^{n+1} \rightarrow \mathbb{R}^{n}$, with domain $D_{i} = \left\{ (x_{0}\cdots x_{n}) \in \mathbb{R}^{n+1}|x_{i}\neq 0)\right\}$ and $f(x_{0}\cdots x_{n}) = (\frac{x_{0}}{x_{i}},\cdots ,\frac{x_{i-1}}{x_{i}},\frac{x_{i+1}}{x_{i}},\cdots \frac{x_{n}}{x_{i}})$ is continuous?