Continuity Checker

Applet to investigate whether a given function is continuous at a point or uniformly continuous^(*).

Continuity at a point: f(x) is continuous at the point x₀ if for every

> 0 there exists a

> 0 such that:

whenever |x - x₀| < then |f(x) - f(x₀)| <

Uniform Continuity: f(x) is uniformly continuous if for every

> 0 there exists a

> 0 such that:

whenever |x - y| < then |f(x) - f(y)| <

These concepts are difficult to understand, but the applet intends to help.

	For a function to be continuous, you must pick an x (by clicking one the graph) and then ensure that the "red" area (where \|x - x₀\| < ) touches the graph of the function inside the "green" area (where \|f(x) - f(x₀)\| < ). Your can achieve that by finding appropriate numbers for each given . If you succeed, the function is continuous at the point.
	For a function to be uniform continuous, you must find a for any given such that the "red" area (where \|x - y\| < ) touches the graph of the function inside the "green" area (\|f(x) - f(y)\| < ) regardless of where shift your focus.

Start the applet, bring up the Options dialog, select "Simple" or "Uniform" continuity, and give it a try.

To enter your own function, use the operators, functions, and constants shown on the right. In particular, the if statement can be used to enter piecewise defined functions.

Operators:
+, -, *, /, ^
Functions:
sin, cos, tan, ln, log, abs, int, frac, asin, acos, atan, sinh, cosh, tanh, asinh, acosh, atanh, ceil, floor, round, exp, sqr, sqrt, sign, fact
Other:
(, ), <, >, <=, >=, and, or
Conditionals:
if(test,if_true,if_false)
Constants
pi, e

Created by Bert Wachsmuth as part of Interactive Real Analysis