If f is continuous and f $\in$ L1 then $\lim_{\tau\to \tau_0} \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $0$?

Question

Where $ \Vert f_\tau-f_{\tau_0}\Vert_{L_1}$ = $\int_\mathbb{T} \vert f(t-\tau)-f(t-\tau_0)\vert \ dt$

I find it easy to see when f is uniformly continuous, since we would have

$\vert f_\tau-f_{\tau_0}\vert < \epsilon$ if we set $\tau$ close enough to $\tau_0$

and then $\Vert f_\tau-f_{\tau_0}\Vert_{L_1} = \int_\mathbb{T}\vert f_\tau(t)-f_{\tau_0}(t)\vert \ dt$ $~$< $~$$\int_\mathbb{T}\epsilon\ d t$ = 2$\pi\epsilon$

But I can't get to this by assuming f is only continuous

Any suggestions please?