Definition of continuous function :

A function said to be a continuous if and only if it satisfy the three conditions.

1. has to be defined at .

2. The limit of the equation has to exist .  ( Left hand side limit and right hand side limit are equal )

3. The value of this limit must equal .

Consider a continuous function  on the open interval .

Graph the function over entire interval.

Observe the graph , the graph consist of absolute minimum when we consider the entire interval.

Graph the function over .

Observe the graph, the graph does not consist of absolute minimum when we consider the interval .

Therefore a continuous function on open interval may not have maximum or minimum.

A continuous function on open interval may not have maximum or minimum.