A problem with a Borel set, a continuous function with a fixed point and a non-null fixed set

Question

Let $\mu$ be the Lebesgue measure on $[0,1]$ and $A$ be a Borel subset of $[0,1]$.
Consider a point $x_0 \in A$ such that for any neighborhood $V$ of $x_0$ in $[0,1]$, we have $\mu(V \cap A)>0$.
Let $f: [0,1] \rightarrow [0,1]$ be a continuous function such that $f(x_0) = x_0$.

Question: Is it true that $\mu(f(A) \cap A)>0$ ?

Remark: If necessary, we can assume that $f$ is $C^1$, or even $C^{\infty}$.

Answer 1

This is wrong, because there is a counterexample (using a $C^\infty$ function $f$):

For example, you can choose $x_0=1/2, f(x) = 2x(1-x), A=[1/2,1]$.

Answer 2

Take $f(x)=x_0$ for all $x$, $A=[0,1]$ to get a counter example.