What is wrong with this proof? (lipschitz functions)

In summary, you cannot simply assume that f is differentiable almost everywhere in [0,1] and invoke the mean value theorem. You need to show that the function is differentiable at all points in the interval. This can be done by showing that the integral of |f'(x)| over [0,1] is finite, which is a property of absolutely continuous functions. Once you have established the differentiability of f, then you can use the mean value theorem to prove that f is Lipschitz.
  • #1
dreamtheater
10
0
Suppose f:[0,1]->R is absolutely continuous and |f'(x)|<M almost everywhere in [0,1]. Prove that f is Lipschitz.

I wrote up the following proof and got significant deductions:

By the Mean Value Theorem, for all x,y in [0,1], there exists c between x and y such that |f(x)-f(y)/(x-y)|=|f'(c)|<M.

That is, for all x,y in [0,1], |f(x)-f(y)/(x-y)|<M.
Then |f(x)-f(y)|<M|x-y|, thus f is Lipschitz.
QED

What is wrong with my proof? (Since f is AC, I know that f is differentiable almost everywhere in [0,1]. Is that not enough to invoke the mean value theorem? If not, how do I prove this theorem?)

Thank you.
 
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  • #2
dreamtheater said:
Suppose f:[0,1]->R is absolutely continuous and |f'(x)|<M almost everywhere in [0,1]. Prove that f is Lipschitz.

I wrote up the following proof and got significant deductions:

By the Mean Value Theorem, for all x,y in [0,1], there exists c between x and y such that |f(x)-f(y)/(x-y)|=|f'(c)|<M.

That is, for all x,y in [0,1], |f(x)-f(y)/(x-y)|<M.
Then |f(x)-f(y)|<M|x-y|, thus f is Lipschitz.
QED

What is wrong with my proof? (Since f is AC, I know that f is differentiable almost everywhere in [0,1]. Is that not enough to invoke the mean value theorem?
No, it is not. For example, the function [itex]f(x)= x^2[/itex] is differentiable everywhere in [0, 1] and the mean value theorem says there must be some c in [0, 1] such that f'(c)= (f(1)- f(0))/(1- 0)= 1. Obviously, that point is x= 1/2. But [itex]g(x)= x^2[/itex] for all x except 1/2 and g(1/2)= 0 is differentiable "almost everywhere" (everywhere except x= 1/2 and a singleton set has measure 0) but there is no c such that g'(c)= (g(1)- g(0))/(1- 0)= 1.

If not, how do I prove this theorem?)

Thank you.
 

1. What is a Lipschitz function?

A Lipschitz function is a type of mathematical function that satisfies a Lipschitz condition. This means that the function's rate of change (or slope) is limited by a constant value. In other words, the function cannot have a steep slope or sharp turns. It is named after the mathematician Rudolf Lipschitz.

2. How do you determine if a function is Lipschitz?

To determine if a function is Lipschitz, you need to calculate its Lipschitz constant. This is done by finding the maximum value of the ratio between the difference in the function's output values divided by the difference in its input values. If this ratio is less than or equal to a constant value, then the function is Lipschitz.

3. Why are Lipschitz functions important?

Lipschitz functions are important in mathematics and science because they have well-behaved properties that make them easier to work with. They are used in various fields, such as calculus, differential equations, and optimization, to solve problems and model real-world phenomena.

4. What are some examples of Lipschitz functions?

Some common examples of Lipschitz functions include linear functions, polynomials, trigonometric functions, and exponential functions. These functions have a finite and constant rate of change, making them Lipschitz. However, not all functions are Lipschitz, such as functions with sharp corners or vertical asymptotes.

5. What are the limitations of Lipschitz functions?

While Lipschitz functions have many useful properties, they also have some limitations. For example, not all functions in nature or real-world problems can be accurately represented by Lipschitz functions. Additionally, calculating the Lipschitz constant for complex functions can be challenging and time-consuming.

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