Epsilon vs. Delta Game

Given an epsilon, the computer finds a delta for which
0 < |x - a| < delta implies that |f(x) - f(a)| < epsilon. The consequence is that the function is continuous at a, because the limit exists and is f(a).
Instructions: Hold the button down on x^2 and select a function.
Enter a number at the bottom to be the value `a', and then enter a positive value for epsilon. Click on `Find delta'.
Epsilon:
Delta:

This is the largest possible choice for delta, given this epsilon.

A more reasonable delta, which may not be the largest possible, is calculated using a simpler formula. Often, it is easy to verify that it is sufficient:

Note that this value is smaller than the one above.

Choose a function for f(x):

Choose a value for a, the value where the limit is taken:

Suggestions: Pick a function, and a value of a. Try several small values for epsilon.
See if you can see any patterns. The largest value for delta may behave strangely,
but the reasonable delta has an easily identifiable pattern. Try to find out what it is.

Now try keeping epsilon fixed while changing the value for a. For some functions,
the reasonable delta does not change. These functions are called uniformly continuous,
because the choice for delta does not need to depend on a, only on epsilon.
Try all of the functions, and find out which ones are uniformly continuous.

For an unedited explanation see this page