# Lipschitz continuity

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?

## Answers and Replies

fresh_42
Mentor
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}##.