AMC 10 2021-fall-a Problem 16

The graph of $$f(x) = |\lfloor x \rfloor| - |\lfloor 1 - x \rfloor|$$ is symmetric about which of the following? (Here $\lfloor x \rfloor$ is the greatest integer not exceeding $x$.) $\textbf{(A) }\text{the }y\text{-axis} \textbf{(B) }\text{the line }x = 1 \textbf{(C) }\text{the origin} \textbf{(D) }\text{ the point }\left(\dfrac12, 0\right) \textbf{(E) }\text{the point }(1,0)$