Problem 7 - Calculus On Manifolds: A Modern Approach To Classical Theorems Of Advanced Calculus

Question

2-7. Let \(f: \mathbf{R}^{n} \rightarrow \mathbf{R}\) be a function such that \(|f(x)| \leq|x|^{2}\). Show that \(f\) is differentiable at 0 .

Step-by-Step Solution

Answer

Thus, \( f \) is differentiable at 0.

Step 1 - Definition of Differentiability

To show that a function is differentiable at a point, we need to show that the limit \[ \lim_{{h \to 0}} \frac{{f(h) - f(0) - abla f(0) \cdot h}}{|h|} = 0 \]holds.

Step 2 - Evaluate at Zero

First, evaluate the function at zero. Since \( f: \mathbf{R}^n \rightarrow \mathbf{R} \) and \(|f(x)| \leq |x|^2\), we have \( f(0) = 0 \).

Step 3 - Boundary Behavior

Given that \(|f(x)| \leq |x|^2\), we can use this to bound \(f(h)\) as \(|f(h)| \leq |h|^2\).

Step 4 - Consider the Differentiability Limit

To check the differentiability limit, notice that\[ \left| \frac{{f(h) - f(0) - abla f(0) \cdot h}}{|h|} \right| = \left| \frac{f(h)}{|h|} \right| \].By the bound, we have \( \left| \frac{f(h)}{|h|} \right| \leq \frac{|h|^2}{|h|} = |h| \).

Step 5 - Taking the Limit

Taking the limit as \( h \to 0 \), it is clear that \( |h| \to 0 \). Thus,\[ \lim_{{h \to 0}} \left| \frac{f(h)}{|h|} \right| = 0 \].

Key Concepts

Limit Definition of Differentiability

Differentiability at a point ensures that the function behaves nicely around that point. According to the limit definition of differentiability, we need to show that the limit: \[ \boxed{ \text{lim}_{{h \to 0}} \frac{{f(h) - f(0) - abla f(0) \bullet h}}{|h|} = 0 } \]holds. Here, 𝑓(ℎ) represents the function at point ℎ, 𝑓(0) is the function at zero, and ∇𝑓(0)·h is the gradient of the function at zero multiplied by ℎ. A function being differentiable at a specific point means:

• The derivative exists at that point.
• The function is smooth, without any abrupt changes, at that point.

Understanding the limit definition is crucial because it lays the foundation for many other calculus concepts, like continuity and integrability. If we confirm that the limit stated above is zero, then we can conclude the function is differentiable at that point.

Bounding Functions

In this problem, bounding the function 𝑓(𝑥) is essential. We know that the function satisfies the inequality: \[ |f(x)| \boxed{\text{≤}} |x|^{2} \] This tells us that the absolute value of the function is always less than or equal to the square of 𝑥. Bounding functions helps in:

• Understanding the maximum possible values the function can take.
• Providing useful estimates to solve limits and integrals.

In our case, this bound helps us to analyze the behavior near the origin, especially when checking the differentiability limit. When we have a bound like this, it is easier to manage the terms when we take the limit.

Behavior at Zero

To show differentiability at zero, it is important to evaluate the function at zero. From the given bound \(|f(x)| \boxed{\text{≤}} |x|^{2} \), we get that: \( f(0) \boxed{= 0} \) The behavior of the function at zero is important because it serves as a reference point when we analyze small changes around zero. Knowing the value of the function at zero, we can better understand how the function behaves as it approaches zero. Understanding the behavior at zero leads us to more easily test conditions for differentiability, as shown in this problem. It provides a starting point for our limit calculations.

Epsilon-Delta Definition of Differentiability

Another way to test differentiability is using the Epsilon-Delta definition. This definition states that a function 𝑓 is differentiable at a point a if:

• For every ϵ > 0, there exists a δ > 0 such that for all h,
• if |h| < δ, then
• \( \frac{|f(h) - f(0) - ∇f(0)·h|}{|h|} \boxed{\text{<}} ϵ \).

In simpler terms:

• We want the difference between the actual function value and the linear approximation at zero to be less than ϵ,
• whenever ℎ is within a specific range defined by δ.

This is another perspective on the same problem. Translating our findings into an Epsilon-Delta format can deepen understanding and strengthen foundational concepts in calculus. It rigorously shows that the function behaves nicely and smoothly at zero, confirming that it is indeed differentiable at that point.