Let f : R R be defined by f(x) = x2 for x R. (a)

Question:

Let f : R → R be defined by f(x) = x2 for x ∈ R.
(a) Show that the inverse image f-1(I) of an open interval I := (a, b) is either an open interval, the union of two open intervals, or empty, depending on a and b.
(b) Show that if I is an open interval containing 0, then the direct image f(I) is not open.