Give an example of a function that fails to satisfy the Lipschitz condition at a point of continuity. Definition of the Lipschitz condition:
A function $f: (a,b)\rightarrow\mathbb{R}$ satisfies the Lipschitz condition at $x\in(a,b)$ iff there is $M>0$ and $\epsilon>0$ such that $\lvert x - y\rvert < \epsilon$ and $y\in(a,b)$ imply that $\lvert f(x) - f(y)\rvert\le M\lvert x-y\rvert$. 
Actual question: 
Give an example of a function that fails to satisfy the Lipschitz condition at a point of continuity. If $f$ is differentiable at $x$, prove that $f$ satisfies the Lipschitz condition at $x$.

Hint: If $f$ is differentiable at $x$, then the quantity
$$\left| \frac{f(y) - f(x)}{y - x} \right|$$
must be bounded in a neighborhood of $x$ (why?). How does this relate to the Lipschitz condition?

For the other question, consider a function with a vertical tangent line, such as $x^{1/3}$.
