Location of Roots Theorem
The location of roots theorem is one of the most intutively obvious properties of continuous functions, as it states that if a continuous function attains positive and negative values, it must have a root (i.e. it must pass through 0).
Let be a continuous function such that and . Then there is some such that .
As , is non-empty. Also, as , is bounded
Thus has a least upper bound,
As is continuous at , such that , which contradicts (1).
Also if :
is continuous imples such that , which again contradicts (1) by the Gap lemma.