Problem 14 - Basic Analysis: Introduction to Real Analysis

Question

Suppose \(f: I \rightarrow \mathbb{R}\) is differentiable at \(c \in I .\) Prove there exist numbers a and \(b\) with the property that for every \(\varepsilon>0,\) there is a \(\delta>0,\) such that \(|a+b(x-c)-f(x)| \leq \varepsilon|x-c|,\) whenever \(x \in I\) and \(|x-c|<\delta .\) In other words, show that there exists a function \(g: I \rightarrow \mathbb{R}\) such that \(\lim _{x \rightarrow c} g(x)=0\) and \(|a+b(x-c)-f(x)| \leq|x-c| g(x)\).

Step-by-Step Solution

Answer

Set \(a = f(c)\), \(b = f'(c)\); \(|a+b(x-c)-f(x)| \leq |x-c|g(x)\), with \(g(x) \to 0\) as \(x \to c\).

Step 1: Understand the problem

We are given that the function \(f\) is differentiable at \(c\) and need to show the existence of numbers \(a\) and \(b\) such that \(|a + b(x-c) - f(x)| \leq \varepsilon|x-c|\) holds for any given \(\varepsilon > 0\). We'll prove this by constructing \(a\) and \(b\), showing how \(f\) can be approximated in terms of a linear function plus a term that vanishes as \(x\) approaches \(c\).

Step 2: Use the definition of differentiability

From differentiability, we know \(\lim_{x \to c} \frac{f(x) - f(c)}{x-c} = f'(c)\), where \(f'(c)\) is the derivative at point \(c\). This suggests that locally around \(c\), \(f(x)\) behaves like \(f(c) + f'(c)(x-c)\). This linear approximation is crucial to construct the numbers \(a\) and \(b\).

Step 3: Set the numbers a and b

Let \(a = f(c)\) and \(b = f'(c)\). These values ensure that the linear approximation aligns with the definition of differentiability; thus, we need to show \(|f(c) + f'(c)(x-c) - f(x)| \leq \varepsilon |x-c|\).

Step 4: Simplify the difference

Calculate the expression \(|a + b(x-c) - f(x)|\) which simplifies to \(|f(c) + f'(c)(x-c) - f(x)|\). This can be rewritten using our understanding from the differentiability condition as the limit operation \(|f'(c)(x-c) - (f(x)-f(c))| \leq \varepsilon |x-c|\).

Step 5: Introduce a function g(x)

Since \(|f(x) - f(c) - f'(c)(x-c)|\) can be expressed in terms of an error function that quantitatively behaves as \(\lim_{x \to c} \frac{|f(x) - f(c) - f'(c)(x-c)|}{|x-c|} = 0\), we can define \(g(x) = \frac{|f(x) - f(c) - f'(c)(x-c)|}{|x-c|}\). The condition \(\lim_{x \rightarrow c} g(x) = 0\) holds.

Step 6: Conclude the solution using g(x)

We have \(|f(x) - f(c) - f'(c)(x-c)| = |x-c| g(x)\) with \(\lim_{x \to c} g(x) = 0\). This confirms that \(|a + b(x-c) - f(x)| \leq |x-c| g(x)\) for \(a = f(c)\) and \(b = f'(c)\), satisfying the required condition as \(|x-c| g(x)\) vanishes near \(c\).

Key Concepts

Linear approximation

Linear approximation is a way to approximate any function by using a linear function or a straight line. It's particularly useful to understand how a function behaves near a point. Suppose you have a function, and you're curious about its behavior around a certain point. By using linear approximation, you can create a simple linear model.

• A linear approximation is essentially finding a line tangent to the function at that point.
• Mathematically, it's expressed as a line equation: \( L(x) = f(c) + f'(c)(x-c) \).
• This tells us that near the point \(c\), \(f(x)\) is approximately \( f(c) + f'(c)(x-c) \).

Think of linear approximation as zooming in really close to a curve so that it looks almost like a line. This line gives you a good estimate of the function's value near the particular point.

Derivative

The derivative of a function at a particular point gives you the slope of the tangent line at that point. This is central to understanding how functions behave as the input values change.

• The derivative is written as \(f'(c)\) and represents the rate of change of the function at \(c\).
• For differentiable functions, this slope tells us the function’s sensitivity to changes in input.
• It's computed as \( \lim_{x \to c} \frac{f(x) - f(c)}{x-c} \).

The derivative allows you to understand whether a function is increasing or decreasing at a given point. It serves as the key tool in many fields for optimizing functions, such as finding maximum and minimum values.

Limit

Limits provide a way to understand the behavior of functions as inputs get close to a specific value. They help us make sense of values approach — even if the function itself doesn't reach that value.

• Formally, the limit as \(x\) approaches \(c\) is written as \( \lim_{x \rightarrow c} f(x)\).
• They're a core part of defining a derivative.
• If \( \lim_{x \rightarrow c} g(x) = 0 \), it means the function approaches zero as \(x\) gets closer to \(c\).

Limits are foundational in calculus and are used to handle cases involving infinity or where functions aren't defined. They give precision to concepts like continuity and differentiability.

Error function

In calculus, when approximating a function using a linear approximation, an error term arises. This is called the error function and it measures the difference between the actual function and the linear approximation.

• The error function captures how much the linear approximation deviates from the actual function.
• It is expressed as \( |f(x) - f(c) - f'(c)(x-c)| \).
• By dividing this error by \( |x-c| \), we define a new function \( g(x) = \frac{|f(x) - f(c) - f'(c)(x-c)|}{|x-c|} \).

As \( x \to c \), if \( \lim_{x \to c} g(x) = 0 \), the error becomes negligible. This tells us that the linear approximation becomes more accurate as \( x \) gets closer to \( c \), showing the importance of choosing a nearby point for linear approximations.