Consider the sequence $a_n(x)=\dfrac{2}{\pi}\arctan(nx)$.
$(a_n)$ converges pointwise to $1$ if $x>0$, $-1$ if $x<0$ and $0$ if $x=0$. It does not converge uniformly as the limit function is not even continuous.
Would there be a definition of convergence (maybe in the framework of measures or distributions) with which $(a_n)$ would converge to a set-valued "function" (which would probably be $\{-1\}$ if $x<0$, $[-1,1]$ if $x=0$ and $\{1\}$ if $x>0$)?