Let $\mathbb{D}^n \subseteq \mathbb{R}^n$ be the closed $n$-dimensional unit ball. Suppose we are given an open subset $U \subseteq \mathbb{D}^n$ of full measure in $\mathbb{D}^n$, and a smooth bounded function $f:U \to \mathbb{R}$.
Do there exist smooth functions $f_k:\mathbb{D}^n \to \mathbb{R}$ which converge uniformly to $f$?
Note that that the derivatives of the original $f$ may explode when we approach $\partial U$. In my specific case $U$ has Hausdorff dimension $\le n-1$, but I am not sure if it matters.
As mentioned by Elad $f(x)=\sin(1/x)$ is a counter-example.
Indeed, suppose that $f_k:[-1,1] \to \mathbb{R}$ converges uniformly to $f$ on $[-1,1]\setminus\{0\}$. Then $f_k$ converges uniformly on the closure $\text{cl}([-1,1]\setminus\{0\} )=[-1,1]$. This implies that the limit is a continuous function on $[-1,1]$, which is a contradiction. (as $\sin(1/x)$ cannot be extended to a continuous function on $[-1,1]$).