Editing Extreme value theorem (section)

==Functions to which the theorem does not apply==

The following examples show why the function domain must be closed and bounded in order for the theorem to apply.  Each fails to attain a maximum on the given interval.

# <math>f(x)=x </math> defined over <math>[0, \infty)</math> is not bounded from above.
# <math>f(x)= \frac{x}{1+x} </math> defined over <math>[0, \infty)</math> is bounded from below but does not attain its least upper bound <math>1</math>.
# <math>f(x)= \frac{1}{x}</math> defined over <math>(0,1]</math> is not bounded from above.
# <math>f(x) = 1-x</math> defined over <math>(0,1]</math> is bounded but never attains its least upper bound <math>1</math>.

Defining <math>f(0)=0</math> in the last two examples shows that both theorems require continuity on <math>[a,b]</math>.