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Use the "Sequence" characterization of continuity (Theorem 4.3.2 (i) ⇒ (iii)) to prove that the following functions are continuous at the indicated real number co: 12x²-5 (a) f: R→R, f(x) = 22+3°, x0 = 4; (b) g: [0, 1] → R, g(x) = x³ + 2x² - 3x + 4, xo = 1;

Use the "Sequence" characterization of continuity (Theorem 4.3.2 (i) ⇒ (iii)) to prove that the following functions are continuous at the indicated real number co: 12x²-5 (a) f: R→R, f(x) = 22+3°, x0 = 4; (b) g: [0, 1] → R, g(x) = x³ + 2x² - 3x + 4, 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
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**Theorem 4.3.2 (Characterizations of Continuity)** Let \( f : A \to \mathbb{R} \), and let \( c \in A \). The function \( f \) is continuous at \( c \) if and only if any one of the following three conditions is met: (i) For all \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that \( |x - c| < \delta \) (and \( x \in A \)) implies \( |f(x) - f(c)| < \varepsilon \); (ii) For all \( V_\varepsilon(f(c)) \), there exists a \( V_\delta(c) \) with the property that \( x \in V_\delta(c) \) (and \( x \in A \)) implies \( f(x) \in V_\varepsilon(f(c)) \); (iii) For all \( (x_n) \to c \) (with \( x_n \in A \)), it follows that \( f(x_n) \to f(c) \). If \( c \) is a limit point of \( A \), then the above conditions are equivalent to (iv) \( \lim_{x \to c} f(x) = f(c) \).
Transcribed Image Text:**Theorem 4.3.2 (Characterizations of Continuity)** Let \( f : A \to \mathbb{R} \), and let \( c \in A \). The function \( f \) is continuous at \( c \) if and only if any one of the following three conditions is met: (i) For all \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that \( |x - c| < \delta \) (and \( x \in A \)) implies \( |f(x) - f(c)| < \varepsilon \); (ii) For all \( V_\varepsilon(f(c)) \), there exists a \( V_\delta(c) \) with the property that \( x \in V_\delta(c) \) (and \( x \in A \)) implies \( f(x) \in V_\varepsilon(f(c)) \); (iii) For all \( (x_n) \to c \) (with \( x_n \in A \)), it follows that \( f(x_n) \to f(c) \). If \( c \) is a limit point of \( A \), then the above conditions are equivalent to (iv) \( \lim_{x \to c} f(x) = f(c) \).
**Title: Proving Continuity Using the Sequence Characterization** In this section, we will use the "Sequence" characterization of continuity (Theorem 4.3.2 (i) ⇔ (iii)) to prove that certain functions are continuous at specified real numbers \(x_0\). **Problem:** Verify the continuity of the following functions at their respective points \(x_0\): **(a)** Function \( f: \mathbb{R} \to \mathbb{R} \) given by \[ f(x) = \frac{12x^2 - 5}{x^2 + 3} \] at the point \( x_0 = 4 \). **(b)** Function \( g: [0, 1] \to \mathbb{R} \) given by \[ g(x) = x^3 + 2x^2 - 3x + 4 \] at the point \( x_0 = 1 \). **Explanation:** To prove the continuity of these functions at the indicated points, we will employ the sequence characterization of continuity. According to this theorem, a function \( f \) is continuous at a point \( x_0 \) if for every sequence \( (x_n) \) that converges to \( x_0 \), the sequence \( (f(x_n)) \) converges to \( f(x_0) \). This method involves showing that the limit of the function as \( x \) approaches \( x_0 \) equals the value of the function at \( x_0 \). You are encouraged to substitute various sequences \( (x_n) \) approaching \( x_0 \) and verify this property for each function to establish continuity.
Transcribed Image Text:**Title: Proving Continuity Using the Sequence Characterization** In this section, we will use the "Sequence" characterization of continuity (Theorem 4.3.2 (i) ⇔ (iii)) to prove that certain functions are continuous at specified real numbers \(x_0\). **Problem:** Verify the continuity of the following functions at their respective points \(x_0\): **(a)** Function \( f: \mathbb{R} \to \mathbb{R} \) given by \[ f(x) = \frac{12x^2 - 5}{x^2 + 3} \] at the point \( x_0 = 4 \). **(b)** Function \( g: [0, 1] \to \mathbb{R} \) given by \[ g(x) = x^3 + 2x^2 - 3x + 4 \] at the point \( x_0 = 1 \). **Explanation:** To prove the continuity of these functions at the indicated points, we will employ the sequence characterization of continuity. According to this theorem, a function \( f \) is continuous at a point \( x_0 \) if for every sequence \( (x_n) \) that converges to \( x_0 \), the sequence \( (f(x_n)) \) converges to \( f(x_0) \). This method involves showing that the limit of the function as \( x \) approaches \( x_0 \) equals the value of the function at \( x_0 \). You are encouraged to substitute various sequences \( (x_n) \) approaching \( x_0 \) and verify this property for each function to establish continuity.
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