CAT 2022 Functions — Previous Year Question

Let $r$ be a real number and $f(x)=\left\{\begin{array}{cl}2 x-r & \text { if } x \geq r \\ r & \text { if } x\lt r\end{array}\right.$ . Then, the equation $f(x)=f(f(x))$ holds for all real values of $x$ where