Question

(a) Prove: If \(f\) is continuous at \(x_{0}\) and \(\lim _{x \rightarrow x_{0}} f^{\prime}(x)\) exists, then \(f^{\prime}\left(x_{0}\right)\) exists and \(f^{\prime}\) is continuous at \(x_{0}\). (b) Give an example to show that it is necessary to assume in (a) that \(f\) is continuous at \(x_{0}\).

Short answer

#Short Answer# To prove that a continuous function's derivative is also continuous at a point \(x_0\), we begin by expressing the derivative's limit using its definition. Then, we use the given fact that the function is continuous at \(x_0\) and the limit of the function's derivative exists as \(x\) approaches \(x_0\). Next, we manipulate the limit expression to show that the limit exists and is equal to the derivative's value at \(x_0\), proving that the derivative is continuous. For an example highlighting the necessity of continuous functions, consider the piecewise function \(f(x)\) = \(x^2 \sin(\frac{1}{x})\) if \(x \neq 0\) and \(f(x)=0\) if \(x=0\). This function is continuous, but its derivative does not exist at \(x=0\), illustrating that continuity of the function is essential for the existence and continuity of its derivative.

Step-by-step solution

Find the expression for \(f'(x_0)\) using the definition of derivative

Using the definition of derivative, we have: \(f'(x_0) = \lim_{h\rightarrow 0} \frac{f(x_0 + h) - f(x_0)}{h}\)

Find the limit of \(f'(x)\) as \(x\) approaches \(x_0\)

Given that \(\lim_{x\rightarrow x_0}f'(x)\) exists, we represent that limit as: \(L = \lim_{x\rightarrow x_0}f'(x)\)

Deduce that the limit of the difference quotient also exists

Given that the function \(f\) is continuous at \(x_0\) and the limit \(L\) exists, we can manipulate the limit as follows: \(f'(x) = \lim_{h\rightarrow 0} \frac{f(x + h) - f(x)}{h}\) Now, substitute \(x_0\) for \(x\) and \(L\) for \(f'(x_0)\): \(L = \lim_{h\rightarrow 0} \frac{f(x_0 + h) - f(x_0)}{h}\) Since this limit exists and is equal to \(L\), we conclude that \(f'(x_0)\) exists and its value is \(L\).

Prove that \(f'\) is continuous at \(x_0\)

We have already shown that \(f'(x_0)\) exists. Now we must prove that it is continuous at \(x_0\). To do this, we want to show that \(\lim_{x\rightarrow x_0}f'(x)\) exists and is equal to \(f'(x_0)\). Since \(f'(x_0) = L\), and we know that \(\lim_{x\rightarrow x_0}f'(x) = L\), we conclude that \(f'\) is continuous at \(x_0\).

Find an example that shows the necessity of \(f\) being continuous at \(x_0\)

Consider the following function: $f(x) = \begin{cases} x^2 \sin\left(\frac{1}{x}\right) & \text{ for } x\neq 0 \\ 0 & \text{ for } x=0 \end{cases}$ This function is continuous on its domain \(\mathbb{R}\). However, the limit of the derivative at \(x=0\) does not exist since the function oscillates infinitely as \(x\) approaches \(0\). \(f'(x)\) exists for all \(x \neq 0\), but \(f'(0)\) does not exist, meaning that \(f'\) is discontinuous at \(0\). This example shows that we need to assume that \(f\) is continuous at \(x_0\) in order to ensure the existence and continuity of \(f'(x_0)\).

Key concepts

Derivative

The derivative of a function is a fundamental concept in calculus. It measures how a function changes when its input changes. Essentially, it tells us the rate at which the function's output is changing at any given point. The formal definition of the derivative of a function \( f \) at a point \( x_0 \) is given by the limit: \[ f'(x_0) = \lim_{h \rightarrow 0} \frac{f(x_0 + h) - f(x_0)}{h} \] This expression represents the slope of the tangent line to the curve at \( x_0 \). If this limit exists, \( f \) is said to be differentiable at \( x_0 \). Differentiability implies some level of smoothness in the function. A function can have a derivative at a point only if it does not have a sharp corner or discontinuity there. Differentiability is a stricter condition than continuity, as every differentiable function is continuous, but not every continuous function is differentiable. In the exercise, proving that \( f'(x_0) \) exists when \( f \) is continuous at \( x_0 \) and the limit \( \lim_{x \rightarrow x_0} f'(x) \) exists, is about guaranteeing this smoothness at the point \( x_0 \).

Limits in Calculus

Limits are at the heart of calculus, helping us understand the behavior of functions as inputs approach a certain value. The limit of a function \( f(x) \) as \( x \) approaches a value \( x_0 \) is denoted by: \[ \lim_{x \rightarrow x_0} f(x) \] This limit represents the value that \( f(x) \) gets closer to as \( x \) approaches \( x_0 \). Limits help define derivatives and integrals, which are cornerstones of calculus. They allow mathematicians to handle concepts involving infinity and change. In the context of derivatives, limits are used to define the derivative itself. If the limit of the difference quotient exists as \( h \) approaches zero, then the derivative exists. During the exercise, when it's given that \( \lim_{x \rightarrow x_0} f'(x) \) exists, it implies that as \( x \) gets closer to \( x_0 \), the values of \( f'(x) \) approach some finite value \( L \). This concept is crucial for understanding the continuity of the derivative, as it links the behavior of \( f'(x) \) around \( x_0 \) to its behavior directly at that point.

Continuity in Real Analysis

In real analysis, continuity is about ensuring that a function behaves predictably without sudden jumps or breaks. A function \( f \) is continuous at a point \( x_0 \) if the following three conditions are satisfied:

• \( f(x_0) \) is defined
• \( \lim_{x \rightarrow x_0} f(x) \) exists
• \( \lim_{x \rightarrow x_0} f(x) = f(x_0) \)

These conditions ensure that the function smoothly passes through \( x_0 \) without interruption.Continuity is essential in calculus because it often guarantees the existence of other properties, like derivatives. In the exercise, the condition that \( f \) must be continuous at \( x_0 \) is crucial for proving that \( f'(x_0) \) exists and is continuous. Without continuity, the behavior of \( f(x) \) near \( x_0 \) could be unpredictable, which could mean \( f'(x_0) \) doesn't exist or is not continuous. The example in the solution showed a function discontinuous at \( x_0 = 0 \) where the derivative doesn't behave consistently as \( x \) approaches \( 0 \). Thus, continuity ensures that the behavior of \( f \) flows consistently at every point, linking values from both sides of \( x_0 \) with its value at that point.