For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$

Question

For a real valued function $f(x,y)$ on $\mathbb{R}^2$ which is in $L_2$, show that $f(x+ε,y+ε) → f(x,y)$ in $L_2$ when $ε → 0.$

Not sure how to go about this problem. I tried Fubini. But that didn't seem to work well. I tried doing t straight from the definition. It seems the point is that it epsilon is changing the $x$ and $y$ coordinate at the same speed.

Answer 1

Approximate $f$ in $L^2$ by a sequence $f_n$ of smooth compactly supported functions. Take a $\delta>0$. For sufficiently large $n$, one has $||f-f_n||<\delta$ and $||T_\epsilon f-T_\epsilon f_n||<\delta$ for all $\epsilon$, where $T_\epsilon g(x,y)=g(x+\epsilon,y+\epsilon)$. Fix such an $n$ and take $\epsilon$ small enough that $||f_n-T_\epsilon f_n||<\delta$. Then $||f-T_\epsilon f||<3\delta$.

Answer 2

Hint. For $f \in C_c(\mathbb R^2)$ the property follows by using uniform continuity of $f$ and dominated convergence (note that $f$ is bounded on its compact support). Now use that $C_c(\mathbb R^2)$ is dense in $L^2(\mathbb R^2)$.