Understanding Function Continuity on Closed Intervals

A function can indeed be continuous on a closed interval, a concept that is fundamental in calculus and mathematical analysis. This article delves into the definition of continuity, how a function can be continuous on a closed interval, and the importance of this property in various theorems and applications.

Definition of Continuity

A function (f(x)) is continuous at a point (c) if it meets the following three conditions:

(f(c)) is defined. The limit of the function as (x) approaches (c) exists. That is, [lim_{x to c} f(x)text{ exists.}] The limit equals the function value. That is, [lim_{x to c} f(x) f(c)]

Continuity on a Closed Interval

To be continuous on a closed interval ([a, b]), a function must be continuous at every point (x) in the interval ([a, b]).

At the endpoints:

At (a), the function needs to be continuous from the right. That is, [lim_{x to a^ } f(x) f(a)]. At (b), the function needs to be continuous from the left. That is, [lim_{x to b^-} f(x) f(b)].

For example, consider the function (f(x) x^2) on the interval ([1, 3]). It is continuous at every point in this interval, including at the endpoints 1 and 3.

Importance of Continuity

Continuity on closed intervals is particularly important because of the Extreme Value Theorem. This theorem states that if a function is continuous on a closed interval ([a, b]), then it attains a maximum and a minimum value on that interval.

Definition of Continuity Revisited

A function (f) defined over some domain (D subseteq mathbb{R}) with real values is continuous at (a in D) if for every (varepsilon > 0), there exists (delta > 0) such that for (x in D) and (|x - a| , it holds (|f(x) - f(a)| .

A function is continuous on some subset (A) of (D), which could be (D) itself, if it is continuous at every (a in A).

For the definition of limits, if (c) is a limit point of the domain (D) of the function, we say that (lim_{x to c} f(x) l) if for every (varepsilon > 0), there exists (delta > 0) such that for (x in D) and (|x - c| , it holds (|f(x) - l| .

Using these definitions, the limit (lim_{x to 0} sqrt{x} 0) can be easily proved as a valid application.

Conclusion

Understanding the continuity of functions on closed intervals is crucial for many applications in mathematics, physics, and engineering. By adhering to the definitions of continuity and limits, one can ensure that functions behave predictably and consistently within their domains.