In the realm of mathematical analysis, understanding the properties of functions such as boundedness, continuity, and uniform continuity is crucial. One classic example illustrating the distinction between a function that is bounded and continuous but not uniformly continuous on the interval [1, 2] is the function:
Introduction
An excellent example is the function f(x) sinleft(frac{1}{x-1}right). This function will serve to demonstrate the nuanced properties of functions in this interval.
Properties of the Function
Boundedness
The sine function oscillates between -1 and 1 for all real numbers. Therefore, for any x in [1, 2], the value of sinleft(frac{1}{x-1}right) will always be within the interval [-1, 1]. This makes the function f(x) sinleft(frac{1}{x-1}right) bounded on the interval [1, 2].
Continuity
The sine function is continuous everywhere, and the term frac{1}{x-1} is well-defined and continuous on the interval [1, 2]. Hence, the composition f(x) sinleft(frac{1}{x-1}right) is continuous on the interval [1, 2].
Uniform Continuity
To establish that f(x) sinleft(frac{1}{x-1}right) is not uniformly continuous on [1, 2], we need to examine the behavior of the function as x approaches 1. As x gets closer to 1, the term frac{1}{x-1} becomes very large, causing the sine function to oscillate rapidly between -1 and 1. This rapid oscillation indicates a failure of the uniform continuity condition.
Proof of Non-Uniform Continuity
Consider any delta > 0. We can find points x_n and y_n in [1, 2] such that |x_n - y_n| but |f(x_n) - f(y_n)| is close to 2. This violates the uniform continuity condition, which requires that for any epsilon > 0, there exists a delta > 0 such that |x - y| .
Conclusion
Thus, the function f(x) sinleft(frac{1}{x-1}right) is bounded and continuous on the interval [1, 2] but not uniformly continuous. This example highlights the subtle differences between boundedness, continuity, and uniform continuity in real analysis.
Additional Examples
Considering other examples, such as the function f(x) frac{1}{(x-1)(x-2)}, also demonstrates a function that is not uniformly continuous on [1, 2]. Similarly, a polynomial of degree 2 or higher, such as f(x) x^2, on the interval [1, 2] would also not be uniformly continuous.
Shift and Transformation
For completeness, it is worth noting that a function can be shifted or transformed to fit within any desired interval, including shifting the function on the interval [0, 1] and then translating it to [1, 2]. Additionally, the Dirichlet function, which is not continuous anywhere, can provide examples of non-uniform continuity but is more complex in nature.
Understanding these examples helps in comprehending the intricacies of continuous and uniform continuous functions, which are fundamental concepts in real analysis and mathematical modeling.