Interactive Real Analysis
- part of 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
Back
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.
Back
f(x) = 1 if x > 0 and f(x) = -1 if x
0
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.