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Use the e-6 definition of continuity to prove that the following functions are continuous at the indicated real number to: (a) f: R→R, f(x) = 7x-2, at xo = 4;

Use the e-6 definition of continuity to prove that the following functions are continuous at the indicated real number to: (a) f: R→R, f(x) = 7x-2, at xo = 4;

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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**Definition 4.3.1 (Continuity):** A function \( f : A \to \mathbb{R} \) is *continuous at a point* \( c \in A \) if, for all \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - c| < \delta \) (and \( x \in A \)) it follows that \( |f(x) - f(c)| < \epsilon \). If \( f \) is continuous at every point in the domain \( A \), then we say that \( f \) is *continuous on* \( A \).
Transcribed Image Text:**Definition 4.3.1 (Continuity):** A function \( f : A \to \mathbb{R} \) is *continuous at a point* \( c \in A \) if, for all \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - c| < \delta \) (and \( x \in A \)) it follows that \( |f(x) - f(c)| < \epsilon \). If \( f \) is continuous at every point in the domain \( A \), then we say that \( f \) is *continuous on* \( A \).
**Educational Content: Proving Continuity Using the \(\varepsilon-\delta\) Definition** To demonstrate the continuity of functions using the \(\varepsilon-\delta\) definition, consider the following examples where we prove the continuity of each function at a specified point \(x_0\): **(a) Function \(f\):** - **Function Definition:** \( f : \mathbb{R} \to \mathbb{R}, \quad f(x) = 7x - 2 \) - **Point of Continuity:** \( x_0 = 4 \) **(b) Function \(g\):** - **Function Definition:** \( g : (0, \infty) \to \mathbb{R}, \quad g(x) = \frac{1}{x} \) - **Point of Continuity:** \( x_0 = 10 \) **(c) Function \(h\):** - **Function Definition:** \( h : \mathbb{R} \to \mathbb{R}, \quad h(x) = 3x^2 + 12 \) - **Point of Continuity:** \( x_0 = 1 \) In each case, apply the \(\varepsilon-\delta\) definition of continuity: A function \( f \) is continuous at a point \( x_0 \) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(|x - x_0| < \delta\), it follows that \(|f(x) - f(x_0)| < \varepsilon\). **Guidelines for Proof:** 1. Calculate the value of \( f(x_0) \). 2. Find an expression for \(|f(x) - f(x_0)|\). 3. Manipulate this expression to find a suitable \(\delta\) in terms of \(\varepsilon\). 4. Verify that with this \(\delta\), \(|f(x) - f(x_0)| < \varepsilon\) for \(|x - x_0| < \delta\).
Transcribed Image Text:**Educational Content: Proving Continuity Using the \(\varepsilon-\delta\) Definition** To demonstrate the continuity of functions using the \(\varepsilon-\delta\) definition, consider the following examples where we prove the continuity of each function at a specified point \(x_0\): **(a) Function \(f\):** - **Function Definition:** \( f : \mathbb{R} \to \mathbb{R}, \quad f(x) = 7x - 2 \) - **Point of Continuity:** \( x_0 = 4 \) **(b) Function \(g\):** - **Function Definition:** \( g : (0, \infty) \to \mathbb{R}, \quad g(x) = \frac{1}{x} \) - **Point of Continuity:** \( x_0 = 10 \) **(c) Function \(h\):** - **Function Definition:** \( h : \mathbb{R} \to \mathbb{R}, \quad h(x) = 3x^2 + 12 \) - **Point of Continuity:** \( x_0 = 1 \) In each case, apply the \(\varepsilon-\delta\) definition of continuity: A function \( f \) is continuous at a point \( x_0 \) if for every \(\varepsilon > 0\), there exists a \(\delta > 0\) such that whenever \(|x - x_0| < \delta\), it follows that \(|f(x) - f(x_0)| < \varepsilon\). **Guidelines for Proof:** 1. Calculate the value of \( f(x_0) \). 2. Find an expression for \(|f(x) - f(x_0)|\). 3. Manipulate this expression to find a suitable \(\delta\) in terms of \(\varepsilon\). 4. Verify that with this \(\delta\), \(|f(x) - f(x_0)| < \varepsilon\) for \(|x - x_0| < \delta\).
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