Are There Functions That Are Uniformly Continuous But Not H¨older Continuous?
In mathematical analysis, the concepts of uniform continuity and H¨older continuity play crucial roles in understanding the behavior of functions. While all H¨older continuous functions are uniformly continuous, the converse is not necessarily true. This article delves into the distinction between these two types of continuity using a classic example.
Understanding Continuity Definitions
To appreciate the difference between uniformly continuous and H¨older continuous functions, it is essential to understand the definitions of each:
Uniform Continuity
A function f: [a, b] to mathbb{R} is uniformly continuous on the interval [a, b] if for every positive value of ε, there exists a positive value of δ such that for all x, y in [a, b], if |x - y|
H¨older Continuity
A function f: [a, b] to mathbb{R} is H¨older continuous with exponent α (where 0 if there exists a constant C such that for all x, y in [a, b], the following inequality holds:
|f(x) - f(y)| ≤ C |x - y|^α.
The smaller the value of α, the stricter the conditions on the function. When α 1, the function is Lipschitz continuous, and when α 0, the function is bounded.
Example of Uniformly Continuous but Not H¨older Continuous
A classic example of a function that is uniformly continuous but not H¨older continuous is the function defined by f(x) sqrt{x} on the interval [0, 1].
Uniform Continuity of f(x) sqrt{x}
The function f(x) sqrt{x} is uniformly continuous on [0, 1]. To prove this, we need to show that for any ε > 0, there exists a δ > 0 such that if |x - y|
Using the identity:
|sqrt{x} - sqrt{y}| |(x - y)/(sqrt{x} * sqrt{y})|.
Since x, y are in [0, 1], we have that sqrt{x} * sqrt{y} is bounded away from zero and is at most 2. Therefore, for small enough |x - y|, we can effectively control |sqrt{x} - sqrt{y}|.
Not H¨older Continuous of f(x) sqrt{x}
However, the function f(x) sqrt{x} is not H¨older continuous with any exponent α ≤ 1. To see why, suppose it were H¨older continuous with some α:
|sqrt{x} - sqrt{y}| ≤ C |x - y|^α.
Let y 0 as x approaches 0. Then:
|sqrt{x} - sqrt{0}| |sqrt{x}|
|x - 0| x.
Hence,
|sqrt{x}| ≤ C |x|^α.
Dividing both sides by |x|^α for x > 0 gives:
|x|^(1/2 - α) ≤ C.
This inequality cannot hold as |x| to 0 if α > 1/2. Therefore, the function f(x) sqrt{x} is uniformly continuous but not H¨older continuous with any exponent α ≤ 1/2.
Conclusion
In summary, the function f(x) sqrt{x} is uniformly continuous on [0, 1] but not H¨older continuous for any exponent α > 1/2. This serves as a clear and illustrative example of the difference between uniform continuity and H¨older continuity.