2019 AMC 10B Problem 9

The function f is defined by f(x) = \lfloor|x|\rfloor - |\lfloor x \rfloor|for all real numbers x, where \lfloor r \rfloor denotes the greatest integer less than or equal to the real number r. What is the range of f?

\textbf{(A) } \{-1, 0\}

\textbf{(B) } \text{The set of nonpositive integers}

\textbf{(C) } \{-1, 0, 1\}

\textbf{(D) } \{0\}

\textbf{(E) } \text{The set of nonnegative integers}

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