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(Continuity). A function f AR is continuous at a point c A if, for all e > 0, there exists a d> 0 such that whenever |x- c < 8 (and x A) it follows that f(x)-f(c) < €. If f is continuous at every point in the domain A, then we say that f is continuous on A. Use the e-d definition of continuity to prove that the following functions are continuous at the indicated real number xo: (a) f: R→R, f(x) = 7x-2, at xo = 4; b) g: (0, ∞) → R, g(x) = 1, xo = 10; (c) h: R→R, h(x) = 3x² + 12, xo = 1.

(Continuity). A function f AR is continuous at a point c A if, for all e > 0, there exists a d> 0 such that whenever |x- c < 8 (and x A) it follows that f(x)-f(c) < €. If f is continuous at every point in the domain A, then we say that f is continuous on A. Use the e-d definition of continuity to prove that the following functions are continuous at the indicated real number xo: (a) f: R→R, f(x) = 7x-2, at xo = 4; b) g: (0, ∞) → R, g(x) = 1, xo = 10; (c) h: R→R, h(x) = 3x² + 12, xo = 1.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Definition of Continuity A function \( f : A \to \mathbb{R} \) is **continuous** at a point \( c \in A \) if, for all \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - c| < \delta \) (and \( x \in A \)) it follows that \( |f(x) - f(c)| < \varepsilon \). If \( f \) is continuous at every point in the domain \( A \), then we say that \( f \) is **continuous on** \( A \). ### Application of the \(\varepsilon\)-\(\delta\) Definition Use the \(\varepsilon\)-\(\delta\) definition of continuity to prove that the following functions are continuous at the indicated real number \( x_0 \): (a) \( f : \mathbb{R} \to \mathbb{R}, \, f(x) = 7x - 2 \), at \( x_0 = 4 \); (b) \( g : (0, \infty) \to \mathbb{R}, \, g(x) = \frac{1}{x} \), at \( x_0 = 10 \); (c) \( h : \mathbb{R} \to \mathbb{R}, \, h(x) = 3x^2 + 12 \), at \( x_0 = 1 \).
Transcribed Image Text:### Definition of Continuity A function \( f : A \to \mathbb{R} \) is **continuous** at a point \( c \in A \) if, for all \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( |x - c| < \delta \) (and \( x \in A \)) it follows that \( |f(x) - f(c)| < \varepsilon \). If \( f \) is continuous at every point in the domain \( A \), then we say that \( f \) is **continuous on** \( A \). ### Application of the \(\varepsilon\)-\(\delta\) Definition Use the \(\varepsilon\)-\(\delta\) definition of continuity to prove that the following functions are continuous at the indicated real number \( x_0 \): (a) \( f : \mathbb{R} \to \mathbb{R}, \, f(x) = 7x - 2 \), at \( x_0 = 4 \); (b) \( g : (0, \infty) \to \mathbb{R}, \, g(x) = \frac{1}{x} \), at \( x_0 = 10 \); (c) \( h : \mathbb{R} \to \mathbb{R}, \, h(x) = 3x^2 + 12 \), at \( x_0 = 1 \).
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