means, colloqually, for any required degree of tolerance
, (a positive number usually thought of as very small)
there exists another positive number  (also usually thought 
of as very small) such that whenever x is less than  units 
away from ,  will always be less than  units away from .

To write it concisely,

For any , there exists a 
  such that  whenever .

 will be continuous at  if  and

       

Suppose we have any required tolerance, . Then
we need to find a  such that if ,
then .

 if and only if
 if and only if
 if and only if


So we simply take any  such that 
 and then turn the above 
backwards and we have:

 =>  =>
 => 

Edwin