$$ f(x)=\frac{\sin(x/2)}{x/2}\prod_{k=1}^{\infty}\cos\left(\frac{x}{3^k}\right), \hspace{1em} x\in\mathbb{R}\backslash\{0\} $$ and $f(0)=1$ to achieve continuity. $m$ denotes the Lebesgue measure.
If we guess that $f\in L^1(m)$, then the decay is hard to control. I have tried the approach of truncating the infinite product to finite products on (geometrically expanding) intervals $\{[3^j\pi/2, 3^{j+1}\pi/2]\}_{j\in\mathbb{N}}$, but this approach does not give a finite upper bound on $\|f\|_1$. Similarly, Holder inequality does not give a finite estimate.
Trying the other direction $f\notin L^1(m)$ is also hard.