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 $x$.)
$\textbf{(A) }\text{the }y\text{-axis}\qquad \textbf{(B) }\text{the line }x = 1\qquad \textbf{(C) }\text{the origin}\qquad \textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)\qquad \textbf{(E) }\text{the point }(1,0)$
Note that \[f(1-x)=|\lfloor 1-x\rfloor|-|\lfloor x\rfloor|=-f(x),\]
so $f\left(\frac12+x\right)=-f\left(\frac12-x\right)$.
This means that the graph is symmetric about $\boxed{\textbf{(D) }\text{ the point }\left(\dfrac12, 0\right)}$.