2022 AMC 10B Problem 24

Consider functions f that satisfy |f(x)-f(y)|\leq \frac{1}{2}|x-y| for all real numbers x and y. Of all such functions that also satisfy the equation f(300) = f(900), what is the greatest possible value of f(f(800))-f(f(400))? \textbf{(A)}\ 25 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 100 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 200

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.