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Chapter 7: Problem 2

Prove that \(f\) has a derivative at \(x_{0}\) if and only if $$ D^{+} f\left(x_{0}\right)=D_{+} f\left(x_{0}\right)=D^{-} f\left(x_{0}\right)=D_{-} f\left(x_{0}\right) $$ In that case, \(f^{\prime}\left(x_{0}\right)\) is the common value of the Dini derivates at \(x_{0}\). (We assume that \(f\) is defined in a neighborhood of \(x_{0}\).)

Short Answer

Expert verified
A function \(f\) is differentiable at \(x_0\) if all four Dini derivatives are equal at \(x_0\), with this common value being \(f'(x_0)\).

Step by step solution

01

Understand the Dini Derivatives

The four Dini derivatives at a point \(x_0\) are defined as follows:- \(D^+ f(x_0)\) is the upper right-hand derivative.- \(D_+ f(x_0)\) is the lower right-hand derivative.- \(D^- f(x_0)\) is the upper left-hand derivative.- \(D_- f(x_0)\) is the lower left-hand derivative.These derivatives measure the best approximations of the function's rate of change from the right and left.
02

Condition for Derivative

For a function \(f\) to have a derivative at \(x_0\), the derivative's limit definition must hold.This means that if \(f\) is differentiable at \(x_0\), the left-hand derivative and right-hand derivative must exist and be equal. In Dini derivative terms, this implies:\[ D^+ f(x_0) = D_+ f(x_0) = D^- f(x_0) = D_- f(x_0) = f'(x_0). \]
03

Prove the Forward Implication

Assume that \( f \) is differentiable at \( x_0 \). By the definition of differentiability, the left and right limits of the difference quotient are equal, so the four Dini derivatives equal each other, equating to the derivative \( f'(x_0) \).
04

Prove the Reverse Implication

Assume the condition \( D^+ f(x_0) = D_+ f(x_0) = D^- f(x_0) = D_- f(x_0) \) holds.This means the best approximations of the derivative from both sides agree on the same number, implying the definition of the derivative is satisfied.Thus, \( f \) is differentiable at \( x_0 \) with derivative \( f'(x_0) \) equal to the common value of the Dini derivatives.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dini Derivatives
Dini derivatives are a set of four expressions that capture how a function changes as you approach a point from different directions. They are especially useful in understanding the behavior of functions that might not be neatly described by standard derivatives. Here鈥檚 a brief look at each of the four Dini derivatives:
  • Upper Right-Hand Derivative (\(D^+ f(x_0)\)): This considers the supremum of the slope of secant lines as you approach from the right side of \(x_0\).
  • Lower Right-Hand Derivative (\(D_+ f(x_0)\)): This involves the infimum of the slope of secant lines from the right of \(x_0\).
  • Upper Left-Hand Derivative (\(D^- f(x_0)\)): This works similarly to \(D^+ f(x_0)\) but from the left side.
  • Lower Left-Hand Derivative (\(D_- f(x_0)\)): This is the infimum of the slope of secant lines from the left of \(x_0\).
Each of these derivatives seeks to understand different aspects of how a function "behaves" around \(x_0\) without necessarily abiding by the stricter rules of differentiability. When all four Dini derivatives are equal, it reflects a strong form of agreement on how the function changes around that point.
Differentiability
Differentiability, at a core level, is about smoothness. A function \(f\) is differentiable at a point \(x_0\) if it has a well-defined tangent line at that point. We consider a function to be differentiable at \(x_0\) if the limit of its difference quotient exists:\[f'(x_0) = \lim_{{h \to 0}} \frac{f(x_0 + h) - f(x_0)}{h}\]For a function to be differentiable at a point, its behavior should not include any abrupt changes or "sharp turns". In terms of Dini derivatives, differentiability at \(x_0\) implies that:
  • The function doesn't just match its tangent line when approached from the right or left independently.
  • All Dini derivatives agree, indicating a unified sense of direction and rate of change at \(x_0\).
When \(D^+ f(x_0) = D_+ f(x_0) = D^- f(x_0) = D_- f(x_0)\), the common value is \(f'(x_0)\), confirming smoothness.
Derivative
In calculus, the concept of a derivative is fundamental for understanding how a function changes. The derivative at a given point measures the immediate rate of change of the function as its inputs change. When speaking of the derivative:
  • The derivative \(f'(x_0)\) is the slope of the tangent line to the function at \(x_0\).
  • If a function is smooth and continuous close to \(x_0\), the derivative can be calculated using limits.
  • The value of the derivative gives us information about whether the function is increasing or decreasing at \(x_0\).
In the case of Dini derivatives, when all four equal the same value, this value is actually \(f'(x_0)\). This illustrates that despite their diverse inputs (right or left, upper or lower), they deliver a consistent rate of change. This is why Dini derivatives are particularly valuable: they help ensure and confirm the existence of the standard derivative of a function under specific conditions.
Mathematical Proofs
Mathematical proofs are logical debates with one goal: to confirm whether a statement or theorem is true. When proving statements related to calculus concepts like differentiability and derivatives, having an understanding of proofs is key. Here鈥檚 how proofs typically unfold:
  • **Assumption:** Start by assuming the statement you wish to prove is correct or incorrect. This frames the approach.
  • **Reasoning:** Use logical deductions and prior knowledge (definitions, theorems) to build upon the assumption.
  • **Conclusion:** Arrive at a point where the assumption logically leads to a valid statement or contradiction, confirming or denying the original assertion.
For proving that a function has a derivative at a particular point if and only if its Dini derivatives align, one embarks on a bidirectional exploration:
  • **Forward Direction:** If a function is differentiable at \(x_0\), demonstrate how that implies all four Dini derivatives are equal.
  • **Reverse Direction:** Show how equal Dini derivatives ensure the function is differentiable, thus arriving at the derivative \(f'(x_0)\).
This approach confirms both implications of the theorem, solidifying the understanding of how Dini derivatives connect with standard differentiation.

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Most popular questions from this chapter

What is the ratio \(\frac{|f(J)|}{|J|}\) for the function \(f(x)=x^{2}\) if \(J=[2,2.001], J=[2,2.0001], J=[2,2.00001] ?\)

Give an example of a differentiable function \(f\) such that $$ f^{\prime}\left(x_{0}\right) \neq \lim _{x \rightarrow x_{0}} f^{\prime}(x) $$ Show that if \(f\) is defined in a neighborhood of \(x_{0}\) and \(\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) exists and is finite, then \(f\) is differentiable at \(x_{0}\) and \(f^{\prime}\) is continuous at \(x_{0}\).

(Derived Numbers) The Dini derivates are sometimes called "extreme unilateral derived numbers." Let \(\lambda \in[-\infty, \infty] .\) Then \(\lambda\) is a derived number for \(f\) at \(x_{0}\) if there exists a sequence \(\left\\{x_{k}\right\\}\) with \(\lim _{k \rightarrow \infty} x_{k}=x_{0}\) such that $$ \lambda=\lim _{k \rightarrow \infty} \frac{f\left(x_{k}\right)-f\left(x_{0}\right)}{x_{k}-x_{0}} $$ (a) For the function \(f(x)=\left|x \cos x^{-1}\right|, f(0)=0\), show that every number in the interval \([-1,1]\) is a derived number for \(f\) at \(x=0 .\) Show that the two extreme derived numbers from the right are 0 and 1 , and the two from the left are \(-1\) and 0 . (b) Show that a function has a derivative at a point if and only if all derived numbers at that point coincide. (c) Let \(f: \mathbb{R} \rightarrow \mathbb{R}\) and let \(x_{0} \in \mathbb{R}\). Prove that if \(f\) is continuous on \(\mathbb{R}\), then the set of derived numbers of \(f\) at \(x_{0}\) consists of either one or two closed intervals (that might be degenerate or unbounded). Give examples to illustrate the various possibilities.

Discuss the limiting behavior as \(x \rightarrow 0\) for each of the following functions. (a) \(\frac{1}{x}\) (b) \(\frac{1}{x^{2}}\) (c) \(\frac{1}{\sin x}\) (d) \(\frac{1}{x \sin x^{-1}}\)

The table shown in Figure \(7.3\) gives the values of two functions \(f\) and \(g\) at certain points. Calculate \((f+g)^{\prime}(1),(f g)^{\prime}(1)\) and \((f / g)^{\prime}(1) .\) What can you assert about \((f / g)^{\prime}(3) ?\) Is there enough information to calculate \(f^{\prime \prime}(3) ?\) \begin{tabular}{|c|c|c|c|c|} \hline\(x\) & \(f(x)\) & \(f^{\prime}(x)\) & \(g(x)\) & \(g^{\prime}(x)\) \\ \hline 1 & 3 & 3 & 2 & 2 \\ 2 & 4 & 4 & 4 & 0 \\ 3 & 6 & 1 & 1 & 0 \\ 4 & \(-1\) & 0 & 1 & 1 \\ 5 & 2 & 5 & 3 & 3 \\ \hline \end{tabular} Figure 7.3. Values of \(f\) and \(g\) at several points.
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