TITLE: Lipschitz continuous
QUESTION [8 upvotes]: Let $\delta$ be an interval in $\mathbb{R}$. Recall that a function $f$ is called Lipschitz continuous on $\delta$ with Lipschitz constant $L$ if there holds
 $|f(x) - f(y)| \leq L|x-y|$ for all $x,y$ in $\delta$.
(a) Show that the composition of Lipschitz continuous functions is again Lipschitz continuous.
(b) Is the pointwise maximum of two Lipschitz continuous functions necessarily Lipschitz
continuous?

REPLY [2 votes]: (b) We have $\max (f,g) = (|f - g| + f + g)/2.$ It's very easy to see the sum and difference of Lipschitz functions is Lipschitz, as is any constant times a Lipschitz function. We also know that $|x|$ is Lipschitz on $\mathbb R.$ It follows that both $f-g, f+g$ are Lipschitz, hence so is $|f-g|$ by (a). Thus $|f-g| + f+g$ is Lipschitz, hence so is $(1/2)(|f-g| + f+g)$ and we're done.