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\textbf{\Large MATH 3912 - Assignment 3}
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\begin{questions}

\question
Recall that a function is $Lip(\alpha)$ if $f(x) - f(y)| \le M |x - y|^\alpha$ for all $x$, $y$ in the
function's domain. 
\begin{parts}
\part
Is it true that a function $f$ that is $Lip(1)$ is continuous? Prove it or give a counter-example.
\part
Is it true that a function $f: D \mapsto R$ is continuous implies that $f$ is $Lip(1)$? If not, give a counter-example.
\part
Is it true that a function $f$ that is $Lip(1)$ is uniformly continuous? Prove it or give a counter-example.

\part
Is it true that if $f$ is uniformly continuous it must be $Lip(1)$? If not, give a counter-example 
(you could try $f(x) = x^{1 \over 3}$ in a neighborhood of $0$)

\end{parts}

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\question
Is the function $f(x) = x^2 \in Lip(1)$? How about the function $f(x) = {1 \over x}$ on the interval $(0, 1)$?


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\question
Show that if $f:[a, b] \mapsto R$ is a function that is differentiable and whose derivative is bounded 
on $[a, b]$ is $Lip(1)$.

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\question
Find a function $f:[0, 1] \mapsto R$ that is differentiable but not $Lip(1)$.

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\question
Find a function $f:[0, 1] \mapsto R$ that is $Lip(1)$ but not differentiable.

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\question
Show that if $f \in Lip(2)$  and $f$ is differentiable then $f$ must be constant. 
As a hint, consider the definition of derivative, then use the Lipschitz condition
to conclude something about $f'$.

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\question
Prove that if $f \in Lip(\alpha)$ with $\alpha > 1$ and $f$ differentiable then $f$ must be 
constant. For extra credit, prove this statement when the differentiability condition is
dropped.

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\end{questions}
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