Is f(t,y)=e^{-t}y Lipschitz Continuous in y?

[the_dane]

This is not so much a "Homework" question I am just giving an example to ask about a specific topic.

Homework Statement

Is $f(t,y)=e^{-t}y$ Lipschitz continuous in $y$

Homework Equations

I don't really know what to put here. Here is the definitions:
https://en.wikipedia.org/wiki/Lipschitz_continuity

The Attempt at a Solution

I have found out that I can determine whether a function is Lipschitz continuous by looking at it's derivative $f_y = df/dy$ and see if it is bounded. In my case $f_y=e^{-t}$ is bounded in $(y,f_y)$ plane but is NOT bounded in $(t,f_y)$ plane. My conclusion is that $f(t,y)$ Lipschitz continuous in $y$, right? I don't see why it should matter if $f_y$ is not bounded in $(t,f_y)$ plane. Is statement correct?

 

[fresh_42]

If you say $f(t,y)=e^{-t}y$ is continuous in $y$, then you only regard $f(t,y)$ as a function of $y$. That is as if you asked, whether $g(y)=c \cdot y$ is Lipschitz continuous, and yes, you are right, it is: $|g(y_1)-g(y_2)|=1 \cdot |y_1-y_2| \leq 1 \cdot |y_1-y_2|$. No criterion other than the definition is needed here.

The picture changes if you consider $f(t,y)$ as a function of $t$, or as a function of $(t,y)$. E.g. in the second case we must show
$$
|f(t_1,y_1)-f(t_2,y_2)|=|e^{-t_1}y_1-e^{-t_2}y_2| \leq L\cdot |(t_1,y_1)-(t_2,y_2)|= L\sqrt{(t_1-t_2)^2+(y_1-y_2)^2)}
$$
which I think is not possible, so it's not Lipschitz on $\mathbb{R}^2$. Similar is true in the first case, if we consider $e^{-t}y=f(t,y)=g(t)=c \cdot e^{-t}$.