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Chapter 49: Problem 5

\(f\) sei auf dem Intervall \(\boldsymbol{I}\) stetig, und zu dem Punkt \(x_{0} \in I\) gebe es eine \(\delta\)-Umgebung \(U\) mit folgender Eigenschaft: \(f\) ist auf \(\dot{U} \cap I\) differenzierbar, und es strebt \(f^{\prime}(x) \rightarrow \eta\) fit \(x \rightarrow x_{0}, x \in \dot{U} \cap I\). Dann existiert auch \(f^{\prime}\left(x_{0}\right)\) und ist \(=\eta\).

Short Answer

Expert verified
The derivative \(f'(x_0)\) exists and equals \(\eta\).

Step by step solution

01

Understanding the Given Conditions

We are given a continuous function \(f\) over an interval \(I\), and a point \(x_0 \in I\). There exists a neighborhood \(U\) around \(x_0\) where the function \(f\) is differentiable except possibly at \(x_0\) itself.
02

Neighborhood and Differentiability

The exercise states that within the punctured neighborhood \(\dot{U} \cap I\) (meaning \(U\) excluding \(x_0\)), the function \(f\) is differentiable. Further, as \(x\) approaches \(x_0\) within this neighborhood, the derivative \(f'(x)\) tends towards a limit \(\eta\).
03

Define Left and Right Limit Approaches

To assess the differentiability of \(f\) at \(x_0\), examine the behavior of \(f'(x)\) as \(x\) approaches \(x_0\) from both sides within the neighborhood, ensuring that this derivative value approaches \(\eta\). This condition implies that the derivative from both sides is consistent as \(x\) nears \(x_0\).
04

Applying the Limit Definition

According to the definition of derivative, \(f'(x_0)\) exists if the limit \[\lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0}\] exists. With \(f'(x)\rightarrow \eta\), we conclude that both the left-handed and right-handed derivatives yield \(\eta\) as \(x\) approaches \(x_0\).
05

Conclusion on Differentiability at \(x_0\)

Since both the left and right hand derivatives exist and equal \(\eta\) and \(f\) is continuous at \(x_0\), \(f'(x_0)\) also exists and is equal to \(\eta\). Essentially, \(f'(x_0) = \eta\).

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Key Concepts

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

Differentiability
Differentiability of a function refers to its ability to have a derivative at a given point. The derivative is essentially the slope of the tangent to the function at that specific point. For a function to be differentiable at a point, the limit defining the derivative must exist at that point.
Let's break this down simply:
  • A function is said to be differentiable at a point if it has a defined slope at that point.
  • This slope is determined by the limit of the function's "rate of change" as it approaches the point from both left and right.
In the context of this exercise, we found that as we approach the point \(x_0\), given the conditions, \(f'(x)\) converges to a certain number \(\eta\), proving the existence of \(f'(x_0)\). Hence, at \(x_0\), the function is differentiable, and its derivative equals \(\eta\).
Continuity
Continuity is another critical feature of functions in calculus. A function is continuous at a given point if there are no interruptions or jumps in its graph at that point. When graphed, a continuous function will appear as an unbroken curve or line.
To be more precise:
  • A function \(f(x)\) is continuous at \(x_0\) if \(\lim_{x \to x_0} f(x) = f(x_0)\).
  • Another way to see this is: the function's value at \(x_0\) matches the value that the function's outputs are approaching as \(x\) approaches \(x_0\).
For the exercise, the condition states that \(f\) is continuous over the interval \(I\). This ensures there's no sudden change in the behavior of \(f(x)\) as \(x\) approaches \(x_0\), which is vital for discussing differentiability and limits.
Limits
Limits are foundational in calculus, focusing on the behavior of functions as inputs approach a specific value. Specifically, limits help to determine what value a function approaches as the variable in the function gets closer and closer to a particular point.
Here's how limits work:
  • Mathematically, the limit of \(f(x)\) as \(x\) approaches \(x_0\) is the value \(L\) that \(f(x)\) nears.
  • For \(f\) to be differentiable at \(x_0\), both the left-hand limit and right-hand limit as \(x\) approaches \(x_0\) must exist and be equal.
In the exercise, since \(f'(x)\) approaches a limit \(\eta\) as \(x\) nears \(x_0\), it provides strong evidence that the derivative exists at \(x_0\), and further cements the differentiability of \(f\) at that point.
Neighborhood
In mathematical analysis, a neighborhood around a point includes all points close to that specific point. Consider it as a small vicinity or circle of influence around that point.
In more detail:
  • A \(\delta\)-neighborhood around \(x_0\) involves all points within a distance \(\delta\) from \(x_0\).
  • This neighborhood is used to examine local properties of a function, such as continuity and differentiability, without considering the entire function.
In the context of the given exercise, a punctured neighborhood \(\dot{U}\) was used, meaning a neighborhood around \(x_0\) excluding the point itself. This helped in understanding how the function behaves all around \(x_0\) without being skewed by the singular behavior at \(x_0\) itself. Differentiability on this punctured neighborhood gave deeper insight into the eventual differentiability of the function at \(x_0\).

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Most popular questions from this chapter

\(f\) sei stetig auf dem Intervall \(I, n\)-mal auf dem Inneren \(I\) differenzierbar und ver. schwinde an \(n+1\) Stellen \(x_{0}

Berechne die folgenden Grenzwerte mit Hilfe des Mittelwer tsatzes: a) \(\lim _{n \rightarrow \infty} n(1-\cos (1 / n))\) b) \(\lim _{n \rightarrow \infty}\left(\sqrt[3]{n^{2}+a^{2}}-\sqrt[3]{n^{2}}\right)\) c) \(\lim _{x \rightarrow a}\left(x^{\alpha}-a^{\alpha}\right) /\left(x^{3}-a^{\beta}\right)(a>0, \beta \neq 0)\)

Sei \(0

Sei \(f^{\prime \prime}(x)>0\) f眉r alle \(x \in I\) (also \(f\) streng konvex in \(I\) ) und \(x_{0} \in I\) Dann ist \(f\left(x_{0}\right)+f^{\prime}\left(x_{0}\right)\left(x-x_{0}\right)

\(f\) und \(\mathrm{g}\) seien stetig auf \([a, b]\) und differenzierbar auf \((a, b)\), ferner sei \(f(a)=g(a)\) und \(0 \leqslant f^{\prime}(x)
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