Does the map that sends each $(x,y)\mapsto y\ln|y|$ satisfy the Osgood condition?

Question

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 $$\int_0^l \cfrac{1}{F(t)} dt = +\infty ?$$

Answer 1

I'll consider $0<y_1,y_2<1/e$ 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, $|f(y_1)-f(y_2)|\le f(|y_1-y_2|)$.