## 6.4. Topology and Continuity

### Examples 6.4.2(a):

Let

*f(x) = x*. Show that^{2}*f*is continuous by proving- that the inverse image of an open interval is open.
- that the inverse image of a closed interval is closed.

*f(x) = x*

^{2}- If
*0 < a < b*then the inverse image of the open interval*(a, b)*is*(-, -) (, )*. In particular, the inverse image is open. - If
*a < 0 < b*then the inverse image of the open interval*(a, b)*is*(-, )*, which is again open. - If
*a < b < 0*, then the inverse image of the open interval*(a, b)*is again open (which set is it ?)

But it is now obvious that the inverse image of closed intervals is again a closed set (note that the empty set is both open and closed).

Hence, we have proved that the function *f(x) = x ^{2}* is
continuous, avoiding the tedious epsilon-delta proof.