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Let $s\in [0,1]$. How to show that the map: $F(s)=(1-s)\int_{0,s}tf(t)d\lambda (t)$ is differentiable when $f\in L_2([0,1],\lambda)$?

I tried to calculate the variation $(F(s+h)-F(s)).h^{-1}$ but without result. Also tried Lebesgue theorem of differentiation under integral, but not better. Please help!

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    Are you trying to show that it is differentiable for all $s$? Because that is not true. The $(1-s)$ factor doesn't do much except at $s=1$, and the $t$ factor in the integral doesn't either except at $t=0$. So at any point in the open interval $(0,1)$, the question is basically whether the integral of an $L^2$ function is differentiable, and in general it is not. The best you can say is that it is absolutely continuous (thus differentiable almost everywhere) and $1/2$-Hölder-continuous.2012-10-26

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