Equality of Continuous Functions on Closed and Bounded Intervals

The behavior of continuous functions on a closed and bounded interval is an important topic in mathematical analysis. While it is possible for two continuous functions to be equal on a specific subset of this interval, they may not be equal everywhere within the interval. This article explores the conditions under which two continuous functions can be equal on a closed and bounded interval but not elsewhere, and introduces the concept of analytic functions as a stronger class of functions for ensuring equality.

Introduction to Continuous Functions

Continuous functions play a fundamental role in mathematics, particularly in calculus and real analysis. A function ( f ) defined on a set ( X ) is continuous at a point ( x_0 ) if for every ( epsilon > 0 ), there exists a ( delta > 0 ) such that for all ( x ) in ( X ) with ( |x - x_0|

Continuous Functions on Closed and Bounded Intervals

Let's consider a closed and bounded interval ([a, b]). On such an interval, the functions ( f(x) 0 ) and ( g(x) ) defined as follows are particularly interesting:

Example Functions

Function ( f(x) )

Define ( f(x) 0 ) for all ( x ) in the set of real numbers ( mathbb{R} ).

Function ( g(x) )

Define ( g(x) 0 ) on the closed interval ([-11, 1]), and ( g(x) expleft( frac{1}{(x^2 - 1)} right) ) on the intervals ((- infty, -1)) and ((1, infty)).

Both ( f(x) ) and ( g(x) ) are continuous on ( mathbb{R} ), and they are equal on the interval ([-1, 1]). However, ( g(x) eq f(x) ) for ( x 1 ).

The Equality Theorem for Continuous Functions

The key issue here is understanding why ( f(x) ) and ( g(x) ) are not equal on the entire interval ([-1, 1]). This is where the concept of the closed and bounded interval comes into play. On a closed and bounded interval, the continuous functions ( f(x) ) and ( g(x) ) demonstrate the following behavior:

Equality on a Closed and Bounded Interval

If two continuous functions are identical on a closed and bounded interval, it does not necessarily mean that they are identical everywhere. The functions provided as examples are continuous and equal on the interval ([-1, 1]), but they differ outside this interval. This situation highlights the need for stronger conditions to ensure that two functions are equal everywhere.

Analytic Functions and the Equality Theorem

A more robust condition for ensuring equality of functions is the concept of analyticity. An analytic function is a function that is locally given by a convergent power series. A key property of analytic functions is the identity theorem, which states that if two analytic functions are equal on a compact non-zero region, then they are equal everywhere in their domain.

The Identity Theorem

Mathematically, if ( f ) and ( g ) are analytic functions and there exists a compact non-zero region ( D ) such that ( f(z) g(z) ) for all ( z in D ), then ( f(z) g(z) ) for all ( z ) in the domain of ( f ) and ( g ).

This theorem implies that the identity of two analytic functions on a compact non-zero region guarantees their equality everywhere. This is a stronger condition than mere continuity.

Conclusion

In summary, continuous functions on a closed and bounded interval can be equal on a subset of this interval but not everywhere. For ensuring equality everywhere, the concept of analytic functions and their identity theorem are more appropriate. The example provided with ( f(x) ) and ( g(x) ) illustrates this distinction clearly, emphasizing the importance of analyticity in the context of function equality.

Key Takeaways

Continuous functions on a closed and bounded interval can be equal on a subset but not everywhere. The identity theorem for analytic functions ensures that if two functions are equal on a compact non-zero region, they are equal everywhere in their domain.

By understanding these concepts, we can better analyze and predict the behavior of functions, particularly in the study of real analysis and mathematical modeling.