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

In summary, the conversation discusses whether a given function is Lipschitz continuous in the variable y, and if so, how to determine it. The conclusion is that the function is indeed Lipschitz continuous in y, but not in t or (t,y). This is because the derivative of the function is bounded in the (y,f_y) plane, but not in the (t,f_y) plane. The conversation also mentions the importance of considering the function as a whole, rather than just in terms of one variable.
  • #1
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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?
 
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  • #2
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}##.
 

1. What is Lipschitz continuity?

Lipschitz continuity is a mathematical concept that describes the behavior of a function. A function is considered Lipschitz continuous if there exists a constant value, called the Lipschitz constant, that bounds the difference between the function values at any two points in its domain. In other words, the function cannot change too quickly or too drastically between any two points.

2. How is Lipschitz continuity related to the function f(t,y)=e^{-t}y?

In the function f(t,y)=e^{-t}y, the Lipschitz constant is equal to e^{-t}. This means that the function is Lipschitz continuous in y, as the value of e^{-t} will always bound the difference between any two points in the function's domain. Essentially, as t increases, the function values will not change too quickly in relation to y.

3. What is the importance of Lipschitz continuity in mathematics?

Lipschitz continuity is important in mathematics because it allows us to make predictions and draw conclusions about the behavior of a function. It also helps us to determine the stability and convergence of numerical methods used to solve differential equations, which is important in many scientific and engineering fields.

4. Can a function be Lipschitz continuous in one variable but not in another?

Yes, it is possible for a function to be Lipschitz continuous in one variable but not in another. In the case of f(t,y)=e^{-t}y, the function is Lipschitz continuous in y but not in t, as the Lipschitz constant varies with the value of t. However, it is also possible for a function to be Lipschitz continuous in both variables.

5. How can we determine if a function is Lipschitz continuous?

To determine if a function is Lipschitz continuous, we need to calculate the Lipschitz constant. This can be done by finding the maximum value of the absolute derivative of the function. If the absolute derivative is bounded by a constant value, then the function is Lipschitz continuous. Additionally, some functions may have a known Lipschitz constant, such as f(t,y)=e^{-t}y, which makes it easier to determine Lipschitz continuity.

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