Let F denote the set of all functions f : R → R taking real numbers as input and producing real numbers as output. (For one example, plusone(x) = x + 1 is a function plusone : R → R, so plusone ∈ F.)...

Let F denote the set of all functions f : R → R taking real numbers as input and producing real numbers as output. (For one example, plusone(x) = x + 1 is a function plusone : R → R, so plusone ∈ F.) Are the following propositions true or false? Justify your answers, including a description of the function(s) if they exist.