Continuity at a Point vs. Continuity in a Neighborhood: Understanding Thomae’s Function and Other Counterexamples

Understanding the relationship between the continuity of a function at a specific point and its continuity in a neighborhood of that point is a fundamental concept in real analysis. A common misconception is that if a function is continuous at a particular point, it must also be continuous in a neighborhood around that point. This article explores why this is not always the case by examining key definitions and providing examples, including a detailed analysis of Thomae’s function and other counterexamples.

Definitions and Key Concepts

First, let us define the concepts of continuity at a point and continuity in a neighborhood:

Continuity at a Point

A function fx is continuous at a point c if the following conditions are met:

fc is defined. lim_{x to c} fx exists. lim_{x to c} fx fc.

These conditions ensure that the function approaches the same value from both sides of the point c, thus ensuring continuity.

Continuity in a Neighborhood

A function is continuous in a neighborhood of c if it is continuous at every point within a small interval around c. This implies that the function behaves smoothly and predictably within that interval.

Counterexamples and Analysis

Despite the intuitive belief that continuity at a single point guarantees continuity in a neighborhood, this is not always the case. We will explore this concept through the lens of Thomae’s function and other counterexamples.

Thomae’s Function

Thomae’s function is a famous counterexample that beautifully illustrates the disconnect between continuity at a point and continuity in a neighborhood. Consider the function f: [0, 1] to [0, 1] defined as follows:

fx 1/q if x p/q, where p and q are relatively prime natural numbers, and gcd(p, q) 1. fx 0 if x is irrational.

Let's analyze f at the point x 0 to demonstrate its properties:

f0 1, because 0 can be considered as 0/1, where 0 and 1 are relatively prime. lim_{x to 0} fx 1, because as x approaches 0, the values of fx are dominated by the rational numbers close to 0, which have a limit of 1. lim_{x to 0} fx eq f0, because the function value at 0 is 1, but the limit as x approaches 0 does not approach 1, as there are irrational numbers close to 0 for which the function value is 0.

Thus, f is not continuous at x 0, even though it seems it could be. By exploring other points, we can find that f is continuous at x p if and only if x p is irrational. This function provides a profound counterexample to the idea that continuity at a point implies continuity in a neighborhood.

Other Counterexamples

Let us consider another classic example: a function that is continuous at only one point. Define fx x if x is rational and fx -x if x is irrational. This function, known as the Dirichlet function, is continuous only at x 0 and discontinuous everywhere else. This can be verified as follows:

At x 0, both limits from the left and right are 0, hence lim_{x to 0} fx 0 f0. For any other point x eq 0, the function oscillates wildly between positive and negative values, making it impossible for the limit to exist or equal the function value.

Another interesting counterexample is the definition of g(x) as 1 if the integer part of x is even and 0 if the integer part is odd. Define f(0) 0 and fx x g(1/x). This function is continuous at x 0 but discontinuous in any neighborhood of 0 because:

At x 0, the limit as x to 0 is 0, matching the function value. As x approaches 0 from the right or left, g(1/x) oscillates between 0 and 1, making the limit of fx oscillate as well.

These counterexamples clearly demonstrate that the continuity of a function at a single point does not guarantee its continuity in a neighborhood around that point.

Conclusion

In conclusion, the concepts of continuity at a point and continuity in a neighborhood are distinct and often do not align. Through the analysis of Thomae’s function and other counterexamples, we have provided clear examples that showcase this important distinction in real analysis. Understanding these nuances is crucial for a deeper grasp of function properties and behavior in calculus and analysis.