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Let $f(x,y)=\begin{cases} y\,\ln|y|,&y\not=0\\0,& y=0 \end{cases} $. Does $f$ satisfy Osgood condition ?

if does. How to find a continuous $F$ on $[0,l]$ s.t. $|f(x,y_1)-f(x,y_2)|\le F(|y_1-y_2|)$ and $\int_0^l \cfrac{1}{F(t)} dt = +\infty .$

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