In whatever follows f denotes a differentiable function from R to R. f∘f
denotes the composition of f(x).
<spanclass=′latex−bold′>(a)</span> If f∘f(x)=f(x)∀x∈R then for all x, f′(x)= or f′(f(x))=. Fill in the
blank and justify.
<spanclass=′latex−bold′>(b)</span>Assume that the range of f is of the form (−∞,+∞),[a,∞),(−∞,b],[a,b].
Show that if f∘f=f, then the range of f is R. (Hint: Consider a maximal
element in the range of f)
<spanclass=′latex−bold′>(c)</span> If g satisfies g∘g∘g=g, then g is onto. Prove that g is either strictly increasing or strictly decreasing. Furthermore show that if g is strictly increasing, then g is unique.