## Examples 6.4.2(b):

Let

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

*f(x) = 1*if

*x > 0*and

*f(x) = -1*if

*x 0*

*(-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.