Proving Local Lipschitz Property for Linear Functions on Real Numbers

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Discussion Overview

The discussion centers on proving the local Lipschitz property for the linear function \( f = mx + c \) on the real numbers. Participants explore definitions and implications of Lipschitz continuity, including both local and global properties.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant questions how to prove that \( f = mx + c \) has a local Lipschitz property on \( \mathbb{R} \).
  • Another participant asserts that the function has a global Lipschitz property with constant \( m \).
  • A participant proposes that if \( g \) is differentiable on \( \mathbb{R} \) and \( g'(x) = m \), then if the derivative is bounded, \( g \) is Lipschitz on \( \mathbb{R} \) and any upper bound for \( |g'(x)| = m \) serves as the Lipschitz constant.
  • It is noted that since \( g' \) is continuous on \( \mathbb{R} \), \( g \) is locally Lipschitz.
  • Another participant reiterates the proof of \( f(x) = mx + c \) being Lipschitz by definition, showing the relationship \( |f(x_1) - f(x_2)| = |m||x_1 - x_2| \).
  • One participant raises a question about the definition of a Lipschitz function, specifically regarding the inequality \( |f(x_1) - f(x_2)| \leq m |x_1 - x_2| \).
  • A brief comment states that \( x = y \) trivially implies \( x \leq y \).

Areas of Agreement / Disagreement

Participants express varying views on the definitions and implications of Lipschitz continuity, with some asserting global properties while others focus on local aspects. The discussion remains unresolved regarding the nuances of the definitions and their applications.

Contextual Notes

There are limitations regarding the assumptions made about differentiability and the implications of continuity on Lipschitz properties. The discussion does not resolve the conditions under which the Lipschitz property is applied.

onie mti
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how do i prove that f= mx+c has a local lipschitz property on R
 
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Re: locally lip function

In fact, it has the global Lipschitz property with constant $m$.
 
Re: locally lip function

Evgeny.Makarov said:
In fact, it has the global Lipschitz property with constant $m$.

is it acceptable to say;
suppose that g is differentiable on R.
g'(x)= m
If the derivative is bounded on R, then g is Lip on R and any upper bound for |g'(x)|=m is the lip constant.

and g' is continuous on R hence g is loc lip.
 
Re: locally lip function

onie mti said:
is it acceptable to say;
suppose that g is differentiable on R.
g'(x)= m
If the derivative is bounded on R, then g is Lip on R and any upper bound for |g'(x)|=m is the lip constant.
Yes. Of course, proving that $f(x)=mx+c$ is Lipschitz by definition is also easy:
\[
|f(x_1)-f(x_2)|=|mx_1+c-(mx_2+c)|=|m(x_1-x_2)|=|m||x_1-x_2|.
\]

onie mti said:
and g' is continuous on R hence g is loc lip.
Every Lipschitz function is locally Lipschitz.
 
Re: locally lip function

Evgeny.Makarov said:
Yes. Of course, proving that $f(x)=mx+c$ is Lipschitz by definition is also easy:
\[
|f(x_1)-f(x_2)|=|mx_1+c-(mx_2+c)|=|m(x_1-x_2)|=|m||x_1-x_2|.
\]

Every Lipschitz function is locally Lipschitz.

but doesn't the def of a Lip function say: | f(x_1) - f(x_2)| less than equal m|(x_1) -(x_2)|
 
$x=y$ trivially implies $x\le y$.
 

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