Let $\{f_n\}$ be a sequence of functions that are continuous and lying in $H^k(\mathbb{R}^m)$. Assuming $k>\frac{m}{2}$, and if $f_n \to f$ pointwise, where $f\in H^k(\mathbb{R}^m)$, such that $f$ has points of isolated disconitnuty on a dense set of measure zero. Sobolev embedding says that $f_n$ cannot converge to $f$ under the norm $\|.\|_{H^k(\mathbb{R}^m)}$. I also believe and want to show that $\|f_n\|_{H^k(\mathbb{R}^m)}$ grows unbounded. How can I prove that and also $$\|f_n\|_{H^k(\mathbb{R}^m)} \to \infty$$ Appreciate some suggestions.

PS : By isolated discontinuties, I mean $f$ is such that it can be obtained from a continuous function $g \in H^k(\mathbb{R}^m)$, by changing its values on a dense set of measure zero.