# Locally Lipschitz trig function

• gaganaut
In summary, tan(x) is continuously differentiable everywhere it is defined, and it is locally Lipschitz according to geometric intuition. However, it is not difficult to find a function that is continuous but not locally Lipschitz, such as x^(1/3) on [-1,1].
gaganaut
Would a trig function like $$tan \left(x\right)$$ be locally Lipschitz?

How do we know that, if we know that $$tan \left(x\right)$$ is not continuously differentiable?

Last edited:
tan(x) is continuously differentiable everywhere where it is defined.

And following my geometrical intuition, I would say that it is locally lip****z, and that you would have to try hard to find a function that is continuous but not locally lipschitz!

quasar987 said:
And following my geometrical intuition, I would say (...) you would have to try hard to find a function that is continuous but not locally lipschitz!
Not that hard though, e.g.
$$[-1,1]\to\mathbb{R}$$
$$x\mapsto x^{1/3}$$
is not Lipschitz on any nhbd of zero.

## 1. What is a Locally Lipschitz trig function?

A Locally Lipschitz trig function is a mathematical function that satisfies the Lipschitz condition on a specific interval or region. This means that the function has a bounded rate of change in that specific region, making it continuous and differentiable at every point.

## 2. How is a Locally Lipschitz trig function different from a regular trig function?

A regular trig function may not satisfy the Lipschitz condition, meaning that it may not have a bounded rate of change in certain regions. A Locally Lipschitz trig function, on the other hand, guarantees that the function is continuous and differentiable in that specific region.

## 3. What is the importance of a Locally Lipschitz trig function?

A Locally Lipschitz trig function is important because it allows us to study the behavior of a function in a specific region or interval, rather than looking at the entire domain. This is particularly useful in applications such as optimization and control theory.

## 4. How is the Lipschitz condition related to the continuity of a Locally Lipschitz trig function?

The Lipschitz condition ensures that a Locally Lipschitz trig function is continuous in the specified region. This is because a bounded rate of change implies that the function does not have any sudden jumps or breaks, making it continuous.

## 5. Can a Locally Lipschitz trig function be non-differentiable?

Yes, a Locally Lipschitz trig function can be non-differentiable at certain points. However, it is still continuous and differentiable at most points within the specified region due to the Lipschitz condition. Non-differentiability may occur at points where the function has sharp corners or cusps.

• Calculus and Beyond Homework Help
Replies
22
Views
553
• Calculus
Replies
3
Views
1K
• Math POTW for University Students
Replies
1
Views
506
• Calculus
Replies
1
Views
1K
• Calculus
Replies
1
Views
1K
• General Math
Replies
3
Views
477
• Calculus
Replies
2
Views
1K
• Calculus
Replies
8
Views
458
• General Math
Replies
17
Views
2K
• Calculus
Replies
8
Views
476