• Let f(x) = 1 if x > 0 and f(x) = -1 if x 0. Show that f is not continuous by
  1. finding an open set whose inverse image is not open.
  2. finding a closed set whose inverse image is not closed.

f(x) = 1 if x > 0 and f(x) = -1 if x 0

The inverse image of the set (-2, 0) is the negative real axis, together with the origin. That set is closed. We have found an open set whose inverse image is not open; therefore the function is not continuous.

The inverse image of the set [0,2] is the positive real axis without the origin. That set is open. We have found a closed set whose inverse image is not closed; therefore, the function is not continuous.