Show that $F: (\mathbb{R}^n, \varepsilon) \rightarrow (\mathbb{R}^n, \rho), \quad \mathbf{x} \mapsto \mathbf{x}$ is continuous, where $\varepsilon$ is the Euclidean metric and $\rho(\mathbf{x},\mathbf{y}) = \sqrt{\sum_{i,j=1}^n a_{ij} (x_i-y_i)(x_j-y_j)}$, and $\mathbf{A}$ is a symmetric positive-definite real $n \times n$ matrix.
This feels like it should be rather easier than I'm currently finding it, considering how close in form the metrics are:
$\sqrt{\sum_{i=1}^n (x_i-y_i)^2} < \delta \Rightarrow \sqrt{\sum_{i,j=1}^n a_{ij} (x_i-y_i)(x_j-y_j)} < \mu$