Can the points at which a series of real valued functions converge form a meagre subset of $\mathbb{R}$? Is this possible except for the case where such set of points is just a singleton?

Let $M$ be any meagre subset of $\mathbb{R}$.

Define $f_i(x)$ to be $0$ on $M$ and to be $(-1)^i$ otherwise. Then the $f_i(x)$ only converge on the points of $M$.