If f(f(x)) = -x for all x \in R then f is injective& There does not exist a continuous function s.t f(f(x)) =-x for all x in R. Problem is, you never get such a function. If f(R) is proper subset of R then f(f(R)) will not be R. So, f is bijective. If f s discts at all points, then fof is discts and not x. Let f be cts on an interval. If f is injective on an interval, then continuous injection of an interval is monotone, so it cant be monotonically decreasing. This is not a valid question.