Let $T_X = (X, \mathcal{T}_X)$ and $T_Y = (Y, \mathcal{T}_Y)$ each be a D1106: Topological space.
Let $B_X = (X, \mathcal{B}_X)$ and $B_Y = (Y, \mathcal{B}_Y)$ each be the D1838: Borel measurable space w.r.t. $T_X$ and $T_Y$, respectively.
Let $f : X \to Y$ be a D55: Continuous map with respect to $T_X$ and $T_Y$.

Then $f$ is a D201: Measurable map with respect to $B_X$ and $B_Y$.