$L^2$ norm convergence with continuity assumption

Question

Let $f,f_1,f_2,\ldots\colon\mathbb{R}\rightarrow\mathbb{R}$ be continuous functions in $L^2(\mathbb{R})$. Suppose that $\|f_n-f\|_2\rightarrow 0$ as $n\rightarrow 0$. Is it true that $f_n(x)\rightarrow f(x)$ for almost every $x$?

Without the continuity assumption, I remember this is not true. But what about with continuity?

Answer 1

No, for all $k\in \mathbb N$ and $j=0,\dots,k-1$ you can find a function $f_{k,j}$ such that $f_{k,j}$ is continuous, $f_{k,j}(x) \ge 1$ for $x \in [j/k,(j+1)/k]$ and $\int f^2_{k,j}\le 2/k$. The sequence $f_{1,0},f_{2,0},f_{2,1},f_{3,0},f_{3,1}\dots$ has the desired properties.