Is the extension in this answer well defined and linear?

Question

I ask this question One assumption in the proof of one result of Hahn-Banach theorem. before. But I have trouble understanding the answer. I am not quite sure the extension in the answer is well defined and linear. And base on my understanding, I think the extension g of f is defined by $g(x)=\lim\limits_{i}f(x_i)$ ($x\in M$). But I don't think it is well defined, since for $x\in\overline{M}$, there may exist $x_i\to x$ and $y_k\to x$. And how to make sure $\lim\limits_{i}f(x_i)=\lim\limits_{k}f(y_k)$?