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If $f:[a,b]\times [c,d]\rightarrow \mathbb{R}$ is Lipschitz continuous, is the following function differentiable with a Lipschitz continuous derivative:

$ F(x,y) = \int_a^b f(s,x)f(s,y) ds\,?$

If not, what is the smoothness of $F$?

Here, Lipschitz continuity of $f(x,y)$ means that for any $(x_1,y_1),(x_2,y_2) \in [a,b]\times [c,d]$ we have

$ \left|f(x_1,y_1) - f(x_2,y_2) \right| \leq K \sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}, $

where $K$ is the Lipschitz constant.

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Not necessarily. Let $f(x,y)=g(x)\,h(y)$ where $f\colon[a,b]\to\mathbb{R}$ and $g\colon[c,d]\to\mathbb{R}$ are Lipschitz. Then $$ F(x,y)=\Bigl(\int_a^b(g(s))^2ds\Bigr)\,h(x)\,h(y). $$ If $h$ is not differentiable, then $F$ is not differentiable.

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