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 [math]a[/math] if for any [math]\epsilon[/math] greater than 0 there exist a [math]\delta[/math] greater than 0 such that for all [math]x[/math] between [math]a - \delta[/math] and [math]a + \delta[/math] then [math]f(x)[/math] is between [math]f(a) - \epsilon[/math] and [math]f(a) + \epsilon[/math]. An equation to state this is [math]\left | x - a \right | \lt \delta \implies \left | f(x) - f(a) \right | \lt \epsilon [/math][br][br]In this applet you can choose from many functions with the next button. Next set the desired [math]\epsilon[/math], set the [math]a[/math] point on the [math]x[/math]-axis, then repeatedly click "closer" to make [math]\delta[/math] smaller. The text will indicate when you have found a suitable [math]\delta[/math].[br][br]The brown box shows the boundaries the function should be inside to meet the desired [math]\epsilon[/math]. Also dashed lines show the actual box the function is inside.

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