Intermediate value theorem
Contents
Statement
Full version
Suppose is a continuous function and a closed interval
is contained in the domain of
(in particular, the restriction of
to the interval
is continuous). Then, for any
between the values
and
(see note below), there exists
such that
.
Note: When we say is between
and
, we mean
if
and we mean that
if
.
Short version
Any continuous function on an interval satisfies the intermediate value property.
Caveats
The statement need not be true for a discontinuous function
It is possible for a function having a discontinuity to violate the intermediate value theorem. Below is an example, of the function where
is the signum function and we define it to be zero at 0.
Here, we consider the domain , with
and
, but there is no
satisfying
, even though
.
The function needs to be defined throughout the domain
Consider the function . We have
and
. However, there is no
such that
. The reason the theorem fails is that
is not defined at the point 0, and hence it is not defined on the domain
.