Show that if a $n \times n$ matrix $A$ is invertible, then the function $T: \mathbb{R}^n \to \mathbb{R}^n$ defined by $T(x)=Ax$ is continuous.

Question

Conjecture: If a $n \times n$ matrix $A$ is invertible, then the function $T: \mathbb{R}^n \to \mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ is continuous in the metric space induced by the $2$-norm on $\mathbb{R}^n$.

My Question: Can any one please tell me if the conjecture is true or false? I failed to prove the conjecture, but I couldn't find a counterexample either. Any hint would be greatly appreciated.