Definition 3.2.1. Let /CR be an open interval, let c € 1, let f: 1- {c} → R be a function and let L E R. The number L is the limit of f as x goes to c, written lim f(x) = L., if for each ɛ > 0, there is some 8 > 0 such that x €I- {c} and |x – e| < d imply |f(x) – L| < ɛ. If lim f(x) = L, we also say that f converges to L as x goes to c. If f converges to some real number as x goes to c, we say that lim f(x) exists. A Definition 3.3.1. Let ACR be a set, and let f:A→R be a function. 1. Let cE A. The function f is continuous at c if for each ɛ > 0, there is some 8 > O such that x E A and ļx– c| < 8 imply |f(x) –f(c) < ɛ. The function f is discontinuous at c if f is not continuous at c, in that case we also say that f has a discontinuity at c. 2. The function f is continuous if it is continuous at every number in A. The function f is discontinuous if it is not continuous.
Definition 3.2.1. Let /CR be an open interval, let c € 1, let f: 1- {c} → R be a function and let L E R. The number L is the limit of f as x goes to c, written lim f(x) = L., if for each ɛ > 0, there is some 8 > 0 such that x €I- {c} and |x – e| < d imply |f(x) – L| < ɛ. If lim f(x) = L, we also say that f converges to L as x goes to c. If f converges to some real number as x goes to c, we say that lim f(x) exists. A Definition 3.3.1. Let ACR be a set, and let f:A→R be a function. 1. Let cE A. The function f is continuous at c if for each ɛ > 0, there is some 8 > O such that x E A and ļx– c| < 8 imply |f(x) –f(c) < ɛ. The function f is discontinuous at c if f is not continuous at c, in that case we also say that f has a discontinuity at c. 2. The function f is continuous if it is continuous at every number in A. The function f is discontinuous if it is not continuous.
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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Using mathematical induction, prove two functions, one linear and one quadratic, are continuous at specific points using the formal definition of a limit. *functions not provided*
![**Definition 3.2.1**
Let \( I \subseteq \mathbb{R} \) be an open interval, let \( c \in I \), and let \( f: I - \{c\} \to \mathbb{R} \) be a function. Let \( L \in \mathbb{R} \). The number \( L \) is the **limit** of \( f \) as \( x \) goes to \( c \), written
\[
\lim_{x \to c} f(x) = L,
\]
if for each \( \varepsilon > 0 \), there is some \( \delta > 0 \) such that \( x \in I - \{c\} \) and \( |x - c| < \delta \) imply \( |f(x) - L| < \varepsilon \). If \(\lim_{x \to c} f(x) = L\), we also say that \( f \) **converges** to \( L \) as \( x \) goes to \( c \). If \( f \) converges to some real number as \( x \) goes to \( c \), we say that \(\lim_{x \to c} f(x)\) **exists**. △
**Definition 3.3.1**
Let \( A \subseteq \mathbb{R} \) be a set, and let \( f: A \to \mathbb{R} \) be a function.
1. Let \( c \in A \). The function \( f \) is **continuous** at \( c \) if for each \( \varepsilon > 0 \), there is some \( \delta > 0 \) such that \( x \in A \) and \( |x - c| < \delta \) imply \( |f(x) - f(c)| < \varepsilon \). The function \( f \) is **discontinuous** at \( c \) if it is not continuous at \( c \); in that case, we also say that \( f \) has a **discontinuity** at \( c \).
2. The function \( f \) is **continuous** if it is continuous at every number in \( A \). The function \( f \) is **discontinuous** if it is not continuous](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8975dae8-fbcb-4b49-8e3a-715e0e53990f%2Fac8d1731-bf48-4a07-99bc-b70162a93acc%2Fke8r8om_processed.png&w=3840&q=75)
Transcribed Image Text:**Definition 3.2.1**
Let \( I \subseteq \mathbb{R} \) be an open interval, let \( c \in I \), and let \( f: I - \{c\} \to \mathbb{R} \) be a function. Let \( L \in \mathbb{R} \). The number \( L \) is the **limit** of \( f \) as \( x \) goes to \( c \), written
\[
\lim_{x \to c} f(x) = L,
\]
if for each \( \varepsilon > 0 \), there is some \( \delta > 0 \) such that \( x \in I - \{c\} \) and \( |x - c| < \delta \) imply \( |f(x) - L| < \varepsilon \). If \(\lim_{x \to c} f(x) = L\), we also say that \( f \) **converges** to \( L \) as \( x \) goes to \( c \). If \( f \) converges to some real number as \( x \) goes to \( c \), we say that \(\lim_{x \to c} f(x)\) **exists**. △
**Definition 3.3.1**
Let \( A \subseteq \mathbb{R} \) be a set, and let \( f: A \to \mathbb{R} \) be a function.
1. Let \( c \in A \). The function \( f \) is **continuous** at \( c \) if for each \( \varepsilon > 0 \), there is some \( \delta > 0 \) such that \( x \in A \) and \( |x - c| < \delta \) imply \( |f(x) - f(c)| < \varepsilon \). The function \( f \) is **discontinuous** at \( c \) if it is not continuous at \( c \); in that case, we also say that \( f \) has a **discontinuity** at \( c \).
2. The function \( f \) is **continuous** if it is continuous at every number in \( A \). The function \( f \) is **discontinuous** if it is not continuous
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