Wikipedia states that the definition of convergence in law only requires that the cumulative distribution functions of the sequence of variables converges pointwise to the CDF of the limit variable at points where the CDF of the limit variable is continuous.
If I were to define convergence in law, I think I would define it as:
$X_n$ converges to $X$ in law if for every event (measurable set of the image space) $E$, $\text P(X_n\in E)\to\text P(X\in E)$
The definition on Wikipedia seems to almost verify the above, but if $x$ is any point of discontuinity, we might not have $\text P(X_n<x)\to\text P(X<x)$. Why not just require convergence everywhere?