Limit of continuous function of sequence of $L^2$ functions

Question

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous function and $y_{n}\to y$ in $L^2(\Omega)$ as $n\to\infty$, $y_n,y:\Omega\to\mathbb{R}$, where $\Omega$ open, bounded subset of $\mathbb{R}^n$. Is it true that then $f(y_{n})\to f(y)$?

Answer 1

If you are asking if $f(y_n) \to f(y)$ in $L^{2}$ norm the answer is NO: On $(0,1)$ take $y_n(x)=\sqrt n I_{(0,\frac 1 {n^{2}})}$, $y=0$ and $f(x)=x^{2}$.