Problem 14 - Basic Analysis: Introduction to Real Analysis

Question

Suppose for \(f:[0,1] \rightarrow \mathbb{R}\) we have \(|f(x)-f(y)| \leq K|x-y|\) for all \(x, y\) in \([0,1],\) and \(f(0)=f(1)=0 .\) Prove that \(|f(x)| \leq K / 2\) for all \(x \in[0,1] .\) Further show by example that \(K / 2\) is the best possible, that is, there exists such a continuous function for which \(|f(x)|=K / 2\) for some \(x \in[0,1] .\)

Step-by-Step Solution

Answer

The bound \(|f(x)| \leq K/2\) is derived and optimal; the function \(f(x) = \frac{K}{2} (2x-1)\) achieves this bound.

Step 1: Apply the Given Condition

We start by considering the given inequality \(|f(x) - f(y)| \leq K |x - y|\) for all \(x, y \in [0,1]\). This asks us to bound the difference between function values using a constant \(K\) scaled by the distance between the arguments.

Step 2: Set Specific Values

To make use of the boundary conditions \(f(0) = 0\) and \(f(1) = 0\), choose \(y = 0\) in the given inequality. So, \(|f(x) - f(0)| = |f(x)| \leq K|x - 0| = Kx\).

Step 3: Analyze Another Specific Function Value

Choose \(y = 1\), then \(|f(x) - f(1)| = |f(x)| \leq K|x - 1| = K(1-x)\).

Step 4: Combine Results from Steps 2 and 3

Now we have two inequalities: \(|f(x)| \leq Kx\) and \(|f(x)| \leq K(1-x)\). Therefore, combining them, we get \(|f(x)| \leq \min(Kx, K(1-x))\).

Step 5: Determine Maximum Bound

The minimum value is covered when \(Kx = K(1-x)\), solving for \(x\) when both are equal gives \(x = 1/2\). Substitute \(x = 1/2\) into either inequality gives \(\left| f\left(\frac{1}{2}\right) \right| \leq \frac{K}{2}\). Therefore, \(|f(x)| \leq \frac{K}{2}\) over \([0, 1]\).

Step 6: Example for Achieving Bound

Consider the function \(f(x) = \frac{K}{2} (2x-1)\). This satisfies \(f(0) = f(1) = 0\) and is a piecewise linear function. The Lipschitz condition is \(|f(x) - f(y)| \leq K|x - y|\), which is satisfied as the slope is \(K\). At \(x = 1/2\), \(|f(x)| = \frac{K}{2}\).

Step 7: Conclusion

We have shown both that \(|f(x)| \leq \frac{K}{2}\) is the highest achievable outcome based on the conditions provided, and provided a function that achieves this bound. The bound \(\frac{K}{2}\) cannot be increased further based on these constraints.

Key Concepts

Real Analysis

Real Analysis is a branch of mathematics focused on understanding the behavior of real numbers, sequences, and functions. It dives into analyzing limits, continuity, differentiability, and integration. In the given problem, real analysis is utilized to scrutinize the properties of a function that adheres to specific conditions within the interval \( [0, 1] \).
One key facet of real analysis is studying how functions behave with certain constraints, like continuity or differentiability. This problem is a classic example, requiring an understanding of how a function changes over a closed interval when it meets particular limits and boundary points. In essence, real analysis provides the mathematical toolkit to discuss and prove properties of functions within specified realms.

Lipschitz Continuity

Lipschitz Continuity is a condition that ensures a function does not oscillate too wildly between any two points. A function \( f \) is said to be Lipschitz continuous if there is a constant \( K \) such that for any two points \( x \) and \( y \) within the interval, \( |f(x) - f(y)| \leq K|x - y| \).
This condition implies a controlled growth or change, meaning the function's rate of change is bound by the constant \( K \), ensuring linear sounding changes.
This concept is pivotal to the given problem, as it helps establish that the difference in function values at any two points is predictably bound, preventing erratic behavior. It simplifies understanding and analyzing function behavior significantly, particularly when proving or disproving inequalities.

Piecewise Linear Function

A piecewise linear function is a function composed of straight-line segments. Each linear segment applies over a specific portion of the function's domain. Such functions are often used to approximate more complex behaviors through simple linear rules with definite slope values.
In the context of the problem, a piecewise linear function is utilized to demonstrate how one could construct a simple function, like \( f(x) = \frac{K}{2} (2x-1) \), which satisfies all the problem's prescribed conditions. These functions are straightforward, making them great candidates to showcase sideways relationships or bounding behavior with the flexibility to produce different slopes as defined segments.
The exercise employs these types of functions to illustrate the concept of satisfying a given Lipschitz condition while reaching optimal bounds.

Inequality Boundaries

Inequality boundaries are mathematical tools to set limits on possible values of functions or variables, typically expressed in terms of greater than or less than relationships. Such boundaries help confine the values a function can achieve and are central to many proofs and properties in mathematical analysis.
In the given exercise, inequality boundaries emerge as key components, asserting that the absolute value of \( f(x) \) cannot exceed \( \frac{K}{2} \) for all \( x \) in the interval \( [0, 1] \). The problem leverages inequality boundaries to establish a maximum limit on the function's value spectrum, derived from the Lipschitz condition and other function constraints.
These boundaries are crucial for guaranteeing that functions obey certain rules, providing a safe zone within which solutions and examples can be verified and applied.