2021 Fall AMC 10A 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}\qquad \textbf{(B) }\text{the line }x = 1\qquad \textbf{(C) }\text{the origin}\qquad \textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)\qquad \textbf{(E) }\text{the point }(1,0)

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