sum and product of Lipschitz functions I have the following question from my notes, where $f$ and $g$ are Lipschitz functions on $A ⊂ \Bbb R$.
I'm able to show that the sum $f + g$ is also a Lipschitz function, however I'm stuck on trying to show that if $f,g$ are bounded on $A$, then the product $fg$ is Lipschitz
on $A$.
Also, is there a valid example of a Lipschitz function $f$ on $[0,+∞)$ such that $f^2$ is
not Lipschitz on $[0,+∞)$?

Suppose $|f| \leq M$ and $|g| \leq M$.
Then
$$ |(fg)(x) - (fg)(y)| \leq |f(x)g(x) - f(x)g(y)| + |f(x)g(y) - f(y)g(y)| \leq M(|g(x) - g(y)|+|f(x)-f(y)|)$$
Consider $f(x) = x \, \forall x \in [0,\infty)$ which is Lipchitz on $[0, \infty)$ but $f^2$ is not.
