To solve Part (b), let's break down the problem. Here's a summary of what we are given and what we need to prove:

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Given Information:

1. $f : \mathbb{R} \rightarrow \mathbb{R}$ is a Lipschitz continuous function with Lipschitz constant $C$, meaning that $|f(x) - f(y)| \leq C |x - y|$ for all $x, y \in \mathbb{R}$.

2. $f$ has an inverse $g$, where $g : \mathbb{R} \rightarrow \mathbb{R}$, such that $g \circ f = I \quad \text{and} \quad f \circ g = I$ where $I$ is the identity map.

3. The inverse function $g$ is also Lipschitz continuous with Lipschitz constant $D$, meaning that $|g(u) - g(v)| \leq D |u - v|$ for all $u, v \in \mathbb{R}$.

Goal:

We need to determine the relationship between the constants $C$ and $D$.

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Solution:

To find the relationship, let's start by using the definition of Lipschitz continuity for $f$ and $g$.

Step 1: Using $f$'s Lipschitz Condition

For $f$, we know: $|f(x) - f(y)| \leq C |x - y|$ for all $x, y \in \mathbb{R}$.

Step 2: Using $g$'s Lipschitz Condition

Since $g$ is the inverse of $f$, let $u = f(x)$ and $v = f(y)$ for $x, y \in \mathbb{R}$. Then, $x = g(u)$ and $y = g(v)$. By the Lipschitz condition for $g$, we have: $|g(f(x)) - g(f(y))| \leq D |f(x) - f(y)|$ But since $g(f(x)) = x$ and $g(f(y)) = y$, this simplifies to: $|x - y| \leq D |f(x) - f(y)|$

Step 3: Combining the Two Inequalities

Now, we have two inequalities:

1. From $f$: $|f(x) - f(y)| \leq C |x - y|$.
2. From $g$: $|x - y| \leq D |f(x) - f(y)|$.

Substituting the first inequality into the second, we get: $|x - y| \leq D \cdot C |x - y|$ for all $x, y \in \mathbb{R}$ with $x \neq y$.

Step 4: Analyzing the Inequality

To satisfy this inequality, we must have: $D \cdot C \geq 1$

Conclusion:

The relationship between the Lipschitz constants $C$ and $D$ is: $C \cdot D \geq 1$

This completes the solution for Part (b).

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Would you like a more detailed breakdown of any step or have any questions?

Related Questions

1. How would we interpret the condition $C \cdot D = 1$ geometrically?
2. What would happen if $C \cdot D < 1$?
3. Can we find examples of functions where $C \cdot D = 1$?
4. Is there a specific type of function for which $C = D = 1$?
5. How does Lipschitz continuity ensure the existence of an inverse function?

Tip

Remember, Lipschitz continuity with a constant less than 1 implies strong control over the rate of change, which can restrict how "spread out" values get mapped, making it important in determining inverse mappings.