continuous functions and basis

Given a function $f:X\to Y$ on topological spaces $(X,\tau )$ and $(Y,\sigma )$ where $\mathcal{B}$ is a topological basis for $\sigma$, it is sufficient to check if f is a continuous map for all the basis elements $B\in \mathcal{B}$

In other words if

$$f^{-1}(B)\in \tau \forall B\in \mathcal{B}$$

then f is a continuous function.

Proof

For any open set $U\subset X$ then

$$U=\bigcup _{\alpha }{B}_{\alpha }$$

Therefore,

$$f^{-1}(U)=\bigcup _{\alpha }{f}^{-1}(B_{\alpha })$$

So if all $f^{-1}(B)$ are open then $f^{-1}(U)$ is open