Terminology: What does "less than 1 and 0 integral" mean? (as in "functions with Lipschitz constant less than one and zero integral")

Question

I am working through Applied Analysis by John K. Hunter. Problem 2.7 states:

Prove that the set of Lipschitz continuous functions on $[0, 1]$ with Lipschitz constant less than one and zero integral is compact in $C([0, 1])$.

What does "less than one and zero integral" mean? $|C|<1$? $0\leq C\leq 1$?

Answer 1

I interpret that to mean the constant is less than one, and also the integral of the function from zero to one is zero. There should be a comma there, that would help.

Answer 2

You are parsing the sentence improperly. The intended meaning is to consider the subset of $C([0,1])$ of $f$ such that $\mathrm{Lip}(f) \leq 1$ and $\int f = 0$. Note here we need $\leq 1$ not $< 1$, because clearly you could take $f(x) = -b/2 + b x$ (a line centered at $1/2$ with slope $b$) and send $b \to 1$.