Fix $x \in \mathbb{R}$ and let $I_{[x]}$ be its indicator function. Does anyone know of a sequence of (obviously) discontinuous approximations $g_n$ to $I_{[x]}$ such that

- $g_n$ converge uniformly to $I_{[x]}$ on $\mathbb{R}$,
- $|g_n(y)-I_{[x,x+n^{-1})}(y)|\in (\frac1{2n},\frac1{n}]$? Is this possible?