SCHWEITZER 1954 Problem 3

Is there a real-valued function $Af$ , defined on the space of the functions, continuous on $[0,1]$ , such that $f(x)\leq g(x)$ and $f(x)\not\equiv g(x)$ inply $Af< Ag$ ? Is this also true if the functions $f(x)$ are required to be monotonically increasing (rather than continuous) on $[0,1]$ ? (R.4)