[Solution Library] Let f be defined on an interval I. Suppose that there exists M>0 and α>0 such that |f(x)-f(y)| ≤q M|x-y|^alpha for all x, y ∈ I.

Question: Let $f$ be defined on an interval $I$. Suppose that there exists $M>0$ and $\alpha>0$ such that

\[|f(x)-f(y)| \leq M|x-y|^{\alpha}\]

for all $x, y \in I$. (Such a function is said to satisfy a Lipschitz condition of order $\alpha$ on $I$.)

Prove that $f$ is uniformly continuous on $I$.
If $\alpha>1$, prove that $f$ is constant on $I$. (Hint: First show that $f$ is differentiable on $I$.)
Show by an example that if $\alpha=1$, then $f$ is not necessarily differentiable on $I$.
Prove that if $g$ is differentiable on an interval $I$, and if $g^{\prime}$ is bounded on $I$, then $g$ satisfies a Lipschitz condition of order 1 on $I$.

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Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document

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[Solution Library] Let f be defined on an interval I. Suppose that there exists M>0 and α>0 such that |f(x)-f(y)| ≤q M|x-y|^alpha for all x, y ∈ I.

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