What is wanted : .
The aim is to show that there exists such a for each . ( will depend on but can't depend on ). It can be achieved by taking , and looking for a condition on so that .
An example :
Let's show the continuity of at .
What is wanted : which can be rewritten
What condition has to be respected by so that ? Take the square root of this inequality ( being an increasing function) : and here is the condition. Hence, by taking , (1) is true and we've shown the continuity of the function at 0.