Uniform Continuity of Functions with Bounded Derivatives

In this article, we will explore the relationship between the boundedness of a function's derivative and its uniform continuity. Specifically, we will prove that if the derivative of a function is bounded on an open interval, then the function is uniformly continuous on that interval. We will also discuss the implications of this relationship and provide counterexamples to demonstrate the necessity of certain conditions.

Introduction

The question of uniform continuity often arises in analysis, and the relationship between a function's derivative and its continuity is of paramount importance. In this context, we consider whether a function with a bounded derivative on an open interval is uniformly continuous. We will demonstrate that the answer is affirmative, and we will also explore the limitations of this relationship.

Proof: Bounded Derivative Implies Uniform Continuity

Let f(x): (a, b) → R be differentiable in the interval (a, b) and f'(x) ≤ M for all x in (a, b).

Step 1: Proving Lipschitz Continuity

By the Mean Value Theorem, for any x2, x1 in (a, b), there exists a point c in (x1, x2) such that:

f'(c) (frac{f(x2) - f(x1)}{x2 - x1})

Since f'(x) ≤ M, we have:

(frac{f(x2) - f(x1)}{x2 - x1}) ≤ M

This implies:

f(x2) - f(x1) ≤ M(x2 - x1)

This inequality shows that the function f(x) is Lipschitz continuous with a Lipschitz constant M.

Step 2: Proving Uniform Continuity

Let ε > 0 be any positive number. Define δ (frac{ε}{M}). If |x - y|

|f(x) - f(y)| ≤ M|x - y|

Since δ does not depend on x or y, we have shown that for any ε > 0, there exists a δ > 0 such that |x - y| f(x) is uniformly continuous on (a, b).

Converse: Not Always True

The converse, however, is not true. A function can be uniformly continuous without its derivative being bounded.

Example: The Square Root Function

Consider the function f(x) √x. This function is uniformly continuous. Given any ε > 0, choose δ ε2. Then for |x - y|

|f(x) - f(y)| |(sqrt{x} - sqrt{y})| (frac{|x - y|}{sqrt{x} sqrt{y}}) ≤ (frac{delta}{2sqrt{x}})

Thus, f(x) √x is uniformly continuous, but its derivative is not bounded near x 0.

Counterexample: An Unbounded Derivative

Consider the function f(x) -(frac{1}{2sqrt{x}}). As x approaches 0, this function becomes unbounded. Thus, a function's uniform continuity does not necessarily imply that its derivative is bounded.

Conclusion

We have shown that if the derivative of a function is bounded on an open interval, then the function is uniformly continuous on that interval. This result is, however, not bidirectional. We provided examples to illustrate the limitations and the necessity of conditions involved in such relationships.