This applet demonstrates the concept of a formal definition of a continuous function. The formal definition says that a function is continuous at a point if for any greater than 0 there exist a greater than 0 such that for all between and then is between and . An equation to state this is
In this applet you can choose from many functions with the next button. Next set the desired , set the point on the -axis, then repeatedly click "closer" to make smaller. The text will indicate when you have found a suitable .
The brown box shows the boundaries the function should be inside to meet the desired . Also dashed lines show the actual box the function is inside.

For each function select an value, then set the desired value. Make the variation in smaller by clicking "Closer". You can reset by changing the value.
What happens when you set at a solid circle point? ( you may need to click "closer" one more time to be sure you have a valid value )
What happens when you set near a hollow point?
What happens when you set where the function is not defined?