Problem 3 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(f: \mathbb{R} \rightarrow \mathbb{R}\) is a differentiable function such that \(f^{\prime}\) is a bounded function. Prove \(f\) is a Lipschitz continuous function.

Step-by-Step Solution

Answer

Since \(|f'(x)| \leq M\), \(f\) is Lipschitz continuous with constant \(M\).

Step 1: Understand the Definition of Bounded Derivative

A function is said to have a bounded derivative if there exists some constant \(M > 0\) such that \(|f'(x)| \leq M\) for all \(x \in \mathbb{R}\). This means the rate of change of \(f(x)\) is limited by \(M\) at any point along the real line.

Step 2: Recall the Meaning of Lipschitz Continuity

A function \(f\) is Lipschitz continuous if there exists a constant \(L\geq 0\) such that for all \(x, y \in \mathbb{R}\), \(|f(x) - f(y)| \leq L|x-y|\). The smallest such \(L\) is called the Lipschitz constant.

Step 3: Apply the Mean Value Theorem

The Mean Value Theorem (MVT) states that if \(f\) is differentiable on \((a, b)\), then there exists some \(c \in (a, b)\) such that \(f'(c) = \frac{f(b) - f(a)}{b - a}\). This implies that \(|f(b) - f(a)| = |f'(c)||b-a|\).

Step 4: Use Bounded Derivative to Prove Lipschitz Condition

Since \(|f'(x)| \leq M\) for all \(x\), from MVT, we have \(|f(b) - f(a)| = |f'(c)||b-a| \leq M|b-a|\). This shows that \(f\) is Lipschitz continuous with \(L = M\).

Step 5: Conclude the Result

We have shown that the bounded derivative condition implies the existence of a constant \(M\) such that for any two real numbers \(a\) and \(b\), \(|f(a) - f(b)| \leq M|a-b|\). Thus, \(f\) is a Lipschitz continuous function.

Key Concepts

Bounded Derivative

A bounded derivative means that the derivative, or the instantaneous rate of change, of a function does not exceed a certain constant. This is crucial because it controls how sharply or steeply a function can ascend or descend. To say a function has a bounded derivative is to say there exists a value \( M > 0 \) such that for every point \( x \) in the function's domain, the absolute value of the derivative, \( |f'(x)| \), is less than or equal to \( M \). This implies:

• The function is not changing too rapidly at any point.
• There are no extreme spikes or valleys without limit.
• Its graph is smooth and does not have breaks or sharp turns.

When differentiability combines with a bounded derivative, as in our exercise, it sets the stage for proving Lipschitz continuity. If a function behaves nicely and has a limited rate of change, it is easier to predict and estimate differences between function values, leading us to the concept of Lipschitz continuity.

Mean Value Theorem

The Mean Value Theorem (MVT) is a powerful tool in calculus that connects the derivative of a function with its overall change over an interval. It states that for a function \( f \) which is continuous on a closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), there exists some point \( c \) in \( (a, b) \) such that:\[ f'(c) = \frac{f(b) - f(a)}{b - a}. \]This theorem essentially says that somewhere between any two points \( a \) and \( b \), the function must have a tangent that is parallel to the secant joining them. Key insights include:

• MVT assures us that the average rate of change is realized at some point within the interval.
• This equivalently means that \(|f(b) - f(a)|\) can be understood or estimated using \(|f'(c)|\times|b-a|\).
• It's a cornerstone in proving properties like Lipschitz continuity when combined with a bounded derivative.

The MVT gives us a bridge from knowing something about local behavior (derivative) to understanding the global behavior of the function, an essential concept when proving Lipschitz continuity.

Differentiable Function

A differentiable function is one that has a derivative at every point in its domain. In essence, at each point \( x \), the function \( f \) can be locally approximated by a line (tangent), and this line changes smoothly without jumping or discontinuities. Differentiability entails:

• Continuity - the function has no gaps or jumps.
• Smoothness - the function does not have sharp corners (cusps).
• Predictability - knowing the function's behavior locally through its derivative.

If a function is differentiable, it behaves tenderly and predictably over its domain. This predictability leads to useful applications in analyzing the function using its derivatives, as seen in proving more complex properties like Lipschitz continuity. In our exercise, differentiability ensures that the function and its bounded derivative are well-defined across the domain, setting up the steps needed to confirm the Lipschitz condition.