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?



Uniform approximation of indicator function of a point
Such a sequence $(g\_n)$ does not exist -- if by $[x]$ you mean $\{x\}$ and if you want $|g\_n(y)-I\_{[x,x+n^{-1})}(y)|\in (\frac1{2n},\frac1{n}]$ to hold for all real $y$.


Indeed, then $|g\_n(x+1/(2n))-1|\in (\frac1{2n},\frac1{n}]$, so that $g\_n(x+1/(2n))\to1$ (as $n\to\infty$) and hence
$$\liminf\_n\sup\_{t\in\mathbb R}|g\_n(t)-I\_{\{x\}}(t)|
\ge\liminf\_n|g\_n(x+1/(2n))-I\_{\{x\}}(x+1/(2n))|=1>0.$$
So, $g\_n$ does not converge uniformly to $I\_{\{x\}}$ on $\mathbb R$.


