The result is not necessarily true, consider for $n\geq1$ the circumferences $S_n$ such that $(\frac{1}{n},0)$ and $(\frac{1}{n+1},0)$ form a diameter of $S_n$, and let $L_n$ be $S_n\cap \{y\leq0\}$ and $R_n$ be the reflection of $L_n$ respect to the $y$ axis.

Now consider the open set bounded by the $L_n$, the $R_n$ and the upper semicircunference of center $0$ and radius $1$. Then the function $x\mapsto m(x)$ is not continuous at $p=(0,0.1)$. This is because $m(p)=(0,0.5)$, but for the sequence of points $p_n=(\frac{1}{2}(\frac{1}{n}+\frac{1}{n+1}),0.1)$, we have $m(p_n)=p_n$ for $n\geq2$.

The result is not necessarily true. Consider for $n\geq1$ the open ball $R_n$ with diameter from $(\frac{1}{n},0)$ to $(\frac{1}{n+1},0)$. Let $L_n$ be the reflection of $R_n$ with respect to the $y$ axis.

Now let $A$ be the union of the $L_n$, the $R_n$ and the upper half of the open unit disc. Then the function $x\mapsto m(x)$ is not continuous at $p=(0,0.1)$. This is because $m(p)=(0,0.5)$, but for the sequence of points $p_n=(\frac{1}{2}(\frac{1}{n}+\frac{1}{n+1}),0.1)$, we have $m(p_n)=p_n$ for $n\geq2$.

The result is not necessarily true, consider for $n\geq1$ the circumferences $S_n$ such that $(0,\frac{1}{n})$ and $(0,\frac{1}{n+1})$ form a diameter of $S_n$, and let $L_n$ be $S_n\cap \{y\leq0\}$ and $R_n$ be the reflection of $L_n$ respect to the $y$ axis.

Now consider the open set bounded by the $L_n$, the $R_n$ and the upper semicircunference of center $0$ and radius $1$. Then the function $x\mapsto m(x)$ is not continuous at $p=(0,0.1)$. This is because $m(p)=(0,0.5)$, but for the sequence of points $p_n=(\frac{1}{2}(\frac{1}{n}+\frac{1}{n+1}),0.1)$, we have $m(p_n)=p_n$.

The result is not necessarily true, consider for $n\geq1$ the circumferences $S_n$ such that $(\frac{1}{n},0)$ and $(\frac{1}{n+1},0)$ form a diameter of $S_n$, and let $L_n$ be $S_n\cap \{y\leq0\}$ and $R_n$ be the reflection of $L_n$ respect to the $y$ axis.

Now consider the open set bounded by the $L_n$, the $R_n$ and the upper semicircunference of center $0$ and radius $1$. Then the function $x\mapsto m(x)$ is not continuous at $p=(0,0.1)$. This is because $m(p)=(0,0.5)$, but for the sequence of points $p_n=(\frac{1}{2}(\frac{1}{n}+\frac{1}{n+1}),0.1)$, we have $m(p_n)=p_n$ for $n\geq2$.