Suppose $f \in L^1[a,b]$. Assume that $\lim_{h \to 0} \int_a^b \frac{1}{h}|f(x+h)-f(x)|\,dx =0$. I'm trying to show that there exists a constant $c$ such that $f(x) = c$ for a.e. $x \in (a, b)$. Given that only $f$ is integrable, I tried using dominated convergence theorem on this integral to try to pull the limit into the integral but I don't think that approach works. I'd appreciate any help on how to go about proving this.
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