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I started writing this post wanting to ask a question, but now I think I answered it. Could you please review my proof?

Definition

A continuous function $f$ defined on the interval $I$ is said to be uniformly continuous if for each $ε>0, ∃ δ>0$ s.t. $∀ x,y∈I, |x−y|≤δ ⇒ |f(x)−f(y)|≤ε$.

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Definition A function $f$ defined on a set $I⊆R$ is said to be Lipschitz continuous on $I$ if there exists an $M$ so that $\frac{|f(x)−f(y)|}{|x−y|}≤M$ for all $x$ and $y$ in $I$ such that $x≠y$.

According to Richard Courant's book in Section 1.2, page 43, Uniform Continuity implies Lipschitz Continuity IF $\delta$, (called the "modulus of continuity") is such that $\delta ≤ \epsilon*C $, where C is a constant. How do I prove this?

Trying to prove this, I go back to the definition for Uniform Continuity on an interval $I$, which I give above and I plugged $\delta ≤ \epsilon*C $ into the above definition, and I get the statement that $∀ε>0$ and for all $x,y$ is some closed interval $I$, $|x−y|≤ε*C ⇒ |f(x)−f(y)|≤ε$.

Then I choose $x,y$ such that $0<|x−y|=ε*C$, and I divide both sides of this inequality---> $|f(x)−f(y)|≤ε$, to get this inequality: $\frac{|f(x)−f(y)|}{|x−y|}≤\frac{1}{C}$.

Well? Is it good?

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You have proved the inequality $|f(x)-f(y)| \leq \frac 1 C |x-y|$ only for a particular choice of $x$ and $y$. hat does not prove that $f$ is Lipschitz.

For a correct proof let $x <y$. we can divide the interval $[x,y]$ into subinetrvals $[x_i,x_{i+1}]$of length at most $\delta=\epsilon C$ using at most $[\frac {y-x} {\delta}]+1$ intervals. We then get $|f(y)-f(x)| \leq \sum_i |f(x_{i+1})-f(x_i)| <\epsilon ([\frac {y-x} {\delta}]+1)$. Can you complete the proof now?

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