## 6.4. Topology and Continuity

### Examples 6.4.3:

Is it true that if

*f*is continuous, then the image of an open set is again open ? How about the image of a closed set ?This is true for inverse images but not for images. Consider the example of a parabola, which certainly represents a continuous function:

*f(x) = x*

^{2}*(-1, 1)*is the set

*[0, 1)*. That set is neither open nor closed; in particular, it is not open.

To find a counterexample for images of closed sets, let's look at the following function:

*[0, )*is the set

*(0, 1]*. Therefore we have found a closed set whose image under a continuous function is not closed (nor open).