Translating we may assume $f(0)=f(2022)=0$. Scaling the interval and the factor 2, we have the following problem: suppose $g:[0,1]\rightarrow R$ has $g(0)=g(1)=0$ and $|g(x)-g(y)|\le |x-y|$. Then $|f(x)-f(y)|\le 1/2$ for all $x,y\in [0,1]$. The proof can be found here https://artofproblemsolving.com/community/c7h3282679_lipschitz_condition_imply_bounded_condition where in some places $<$ has to replaced by $\le$.