$L_{x}f=f$ for all $x \in \mathbb{R}$ implies $f=0$

Question

Let $f\in L^{2}(\mathbb{R})$ if $ L_{x}f=f$ for all $x\in \mathbb{R}$ in $L^{2}(\mathbb{R})$ then $f=0$. That is 'if for all $x\in \mathbb{R}$, $ f(y-x)=f(y)$ for almost every $y \in \mathbb{R}$ (we take lebesgue measure) then $f$ is zero almost everywhere'

Is this a true statement? If true what is the proof ? I couldn't figure out a proof, that is why I put it here.