Is it true that $f(X)$ is countable implies $f$ is measurable?

Question

Let $(X, \mathcal{A})$ and $(Y, \mathcal{B})$ be measurable spaces, and $f:X \to Y$. We call $f$ is measurable if and only if $$\forall B \in \mathcal B: f^{-1} (B) \in \mathcal A$$

I would like to ask if $f(X)$ is countable implies $f$ is measurable. Thank you so much!

Answer 1

Take a non measurable subset $A$ of $X$ and consider its indicator function $\chi_A$.