Let $$f(x,y)=\begin{cases} y\,\ln|y|&\text{if }y\not=0,\\0&\text{if }y=0. \end{cases}$$
Does $f$ satisfy the Osgood condition?
If it does, then how can we find a continuous $F$ on $[0,l]$ such that $$|f(x,y_1)-f(x,y_2)|\le F(|y_1-y_2|)$$ and $$\int_0^l \cfrac{1}{F(t)} dt = +\infty ?$$