# Lipschitz continuity

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?

$$|f(t_1,y_1)-f(t_2,y_2)|=|e^{-t_1}y_1-e^{-t_2}y_2| \leq L\cdot |(t_1,y_1)-(t_2,y_2)|= L\sqrt{(t_1-t_2)^2+(y_1-y_2)^2)}$$