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).
Statement
Let be a continuous function such that
and
. Then there is some
such that
.
Proof
Let
As ,
is non-empty. Also, as
,
is bounded
Thus has a least upper bound,
If :
As is continuous at
,
such that
, which contradicts (1).
Also if :
is continuous imples
such that
, which again contradicts (1) by the Gap lemma.
Hence, .