How to show that $f : (0,\infty) \to \mathbb{R}^2$ given by $f(t)=(t , \sin(1/t))$ is continuous using open sets?

Question

How can I prove the function $f : (0,\infty) \to \mathbb{R}^2$ such that $f(t)=\left(t , \sin\left(\frac{1}{t}\right)\right)$ is a continuous function using that $f^{-1}(U)$ is an open set for every $U$ open set in $\Bbb{R}^2$ ?