Lipschitz Continuity of $f(t,y)=e^{-t(t^2+y^2)}-(t^2+y^2)^{\frac{1}{4}}-sin(t)$ Prove that  the $$f(t,y)=e^{-t(t^2+y^2)}-(t^2+y^2)^{\frac{1}{4}}-sin(t)$$ 
where 
$t^2+y^2 \leq 1$ is not Lipschitz continuous with respect to $y$ in any region around $(0,0)$  This $f(t,y)$ is part of an o.d.e problem im trying to solve
$y'=f(t,y)$ t positive ,$y(0)=0.$
So i need to see how $|f(t,y_1)-f(t,y_2)|$ is behaving 
and if that quantity is bounded. I tried to use some kind of meanvalue argument .
that there exists $u$ s.t
$$\frac{f(t,y_1)-f(t,y_2)}{y_1-y_2}=f_y(u)$$ i dont know if i can do that and even if i can i dont get anythng i think.
.

Generally, one proves Lipschitz continuity by taking the derivative with respect to $y$ and showing it is bounded. 
This strategy will fail here because of the term $(t^2+y^2)^{\frac{1}{4}}$, which has unbounded $y$-derivative (and therefore is not Lipschitz continuous) in a neighborhood of $(0,0)$. In fact, this can be seen without derivatives: 
$$f(0,y)=1-|y|^{\frac{1}{2}}$$ 
is not Lipschitz continuous, as
$$
\frac{|f(0, y)-f(0,0)|}{|y-0|} \to \infty,\quad y\to0
$$
