Let $f\in L^1([0,1],\lambda)$ I'd like to show that $F(x)=\int_{[0,x]}|f|\, d\lambda$ is continuous.
I'm thinking of showing it is Lipschitz, but I can't really find any upper bound for $f$. Or maybe I can say something like $|f(x)|\leq \|f\|_1$ almost everywhere?
Any help is welcome...