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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Question
![**Statement:**
Prove that if
\[
\lim_{{h \to 0}} \frac{{f(c + h) - f(c)}}{h}
\]
exists, then \( f \) is continuous at \( c \).
**Explanation:**
The above expression represents the definition of the derivative of \( f \) at the point \( c \). If this limit exists, it implies that the function \( f \) is differentiable at \( c \). It is a fundamental theorem in calculus that differentiability at a point implies continuity at that point.
**Steps:**
1. **Definition of Continuity**: A function \( f \) is continuous at a point \( c \) if
\[
\lim_{{x \to c}} f(x) = f(c)
\]
2. **Using the Derivative Definition**: The limit
\[
\lim_{{h \to 0}} \frac{{f(c + h) - f(c)}}{h}
\]
implies
\[
\lim_{{h \to 0}} (f(c + h) - f(c)) = 0
\]
because the derivative exists and is finite.
3. **Link to Continuity**: From the above, it follows that
\[
\lim_{{h \to 0}} f(c + h) = f(c)
\]
which confirms that \( f \) is continuous at \( c \) based on the definition of continuity.
**Conclusion:**
The existence of the derivative at \( c \) implies that \( f \) is continuous at \( c \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd5936ff2-300d-435d-83da-c517ddce4903%2F98bc69f8-e5f0-4218-849f-417eac658e21%2Fqp92mkz_processed.png&w=3840&q=75)
Transcribed Image Text:**Statement:**
Prove that if
\[
\lim_{{h \to 0}} \frac{{f(c + h) - f(c)}}{h}
\]
exists, then \( f \) is continuous at \( c \).
**Explanation:**
The above expression represents the definition of the derivative of \( f \) at the point \( c \). If this limit exists, it implies that the function \( f \) is differentiable at \( c \). It is a fundamental theorem in calculus that differentiability at a point implies continuity at that point.
**Steps:**
1. **Definition of Continuity**: A function \( f \) is continuous at a point \( c \) if
\[
\lim_{{x \to c}} f(x) = f(c)
\]
2. **Using the Derivative Definition**: The limit
\[
\lim_{{h \to 0}} \frac{{f(c + h) - f(c)}}{h}
\]
implies
\[
\lim_{{h \to 0}} (f(c + h) - f(c)) = 0
\]
because the derivative exists and is finite.
3. **Link to Continuity**: From the above, it follows that
\[
\lim_{{h \to 0}} f(c + h) = f(c)
\]
which confirms that \( f \) is continuous at \( c \) based on the definition of continuity.
**Conclusion:**
The existence of the derivative at \( c \) implies that \( f \) is continuous at \( c \).
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