functions - Do we have always $f(A cap B) = f(A) cap f(B)$?

Suppose $A$ and $B$ are subsets of a topological space and $f$ is any function from $X$ to another topological space $Y$. Do we have always $f(A \cap B) = f(A) \cap f(B)$?

Thanks in advance

Answer

Let $y \in f(A\cap B)$. So there is an $x \in A\cap B$, so $f(x) = y \in f(A\cap B)$. Then obviously $x \in A$, so $y = f(x) \in f(A)$. Also $x \in B$, so $y = f(x) \in f(B)$. This proves that $f(A\cap B) \subseteq f(A) \cap f(B)$.

Now for the other way: as an example, say that $f: \mathbb{R} \to \mathbb{R}$ and $A = [0,1]$ and $B = [2,3]$, can you find both sides for a simple example of $f$?