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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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I'll consider $0 only. Omit the unused argument $x$. The derivative $f'(y)=\ln y+1$ is increasing and negative. Therefore, for $h>0$ such that $y+h<1/e$ the function $f(y+h)-f(y)$ has positive derivative with respect to $y$, i.e., it is an increasing function of $y$.

Conclusion: $f(h)-f(0)\le f(y+h)-f(y)\le 0$. Returning to the original notation, $|f(y_1)-f(y_2)|\le f(|y_1-y_2|)$.