Let $(E,\tau_E)$ and $(G,\tau_G)$ be topological spaces, and define their Borel $\sigma$-algebras by $\mathcal{B}(E)=\sigma(\tau_E)$ and $\mathcal{B}(G)=\sigma(\tau_G)$. If $f:E\to G$ is continuous, then $f:(E,\mathcal{B}(E))\to (G,\mathcal{B}(G))$ is measurable, meaning that $f^{-1}(B)\in\mathcal{B}(E)$ for every $B\in\mathcal{B}(G)$.