Epsilon-delta definition for functions An epsilon-delta definition for a function \(f(x)\) defines the function's behavior around a given point \(c\) by sta...
Epsilon-delta definition for functions
An epsilon-delta definition for a function (f(x)) defines the function's behavior around a given point (c) by stating that the function value is arbitrarily close to (f(c)) whenever the difference between (x) and (c) is less than (\epsilon). In other words, the function is continuous at (c) if the distance between (x) and (c) is sufficiently small.
Formally, the epsilon-delta definition for a function (f(x)) can be written as follows:
(\text{if } 0 < |x - c| < \epsilon, \text{ then } |f(x) - f(c)| < \delta)
where (\epsilon) is a positive number called the epsilon and (\delta) is a positive number called the delta.
Examples:
(f(x) = x^2) is continuous at (c = 0) because the limit of (f(x)) as (x) approaches (0) is (f(0) = 0^2 = 0), which is equal to the value of (f(c)).
(f(x) = \frac{1}{x}) is not continuous at (c = 0) because the limit of (f(x)) as (x) approaches (0) is (f(0) = \infty), which is not equal to the value of (f(c)).
(f(x) = \sqrt{x}) is continuous at (c = 0) because the limit of (f(x)) as (x) approaches (0) is equal to (f(c) = \sqrt{0} = 0), which is equal to the value of (f(c))