This page presents questions on the concept of continuity and continuous functions in calculus, along with their solutions. These questions are designed to help you gain a deep understanding of continuity.
True or False: If a function \( f \) is not defined at \( x = a \), then it is not continuous at \( x = a \).
Answer: True. See the definition of continuous functions.
True or False: If \( f \) is a function such that
\[
\lim_{x \to a} f(x) \text{ does not exist},
\]
then \( f \) is not continuous at \( x = a \).
Answer: True. For \( f \) to be continuous at \( x = a \), \( \lim_{x \to a} f(x) \) must exist and equal \( f(a) \), and \( f \) must be defined at \( x = a \).
True or False: All polynomial functions are continuous.
Answer: True. This follows from the theorem that all polynomials are continuous everywhere.
If functions \( f(x) \) and \( g(x) \) are continuous everywhere, then
(A) \( \frac{f}{g}(x) \) is continuous everywhere.
(B) \( \frac{f}{g}(x) \) is continuous everywhere except at the zeros of \( g(x) \).
(C) More information is needed to answer this question.
Answer: (B). The function \( \frac{f}{g}(x) \) is undefined at zeros of \( g(x) \).
If \( f(x) \) and \( g(x) \) are continuous everywhere and
\[
f(1) = 2, \quad f(3) = -4, \quad f(4) = 8, \quad g(0) = 4, \quad g(3) = -6, \quad g(7) = 0,
\]
find \( \lim_{x \to 3} (f + g)(x) \).
(A) -10
(B) -11
(C) Cannot determine
Answer: (A). Since \( f \) and \( g \) are continuous,
\[
\lim_{x \to 3} (f+g)(x) = \lim_{x \to 3} f(x) + \lim_{x \to 3} g(x) = f(3) + g(3) = -4 + (-6) = -10.
\]
The statement "If \( f(x) = \sin x \), then \( f \) is continuous" is true. Which of the following is also true?
(A) If \( f(x) \neq \sin x \), then \( f \) is not continuous.
(B) If \( f \) is not continuous, then \( f(x) \neq \sin x \).
(C) If \( f \) is continuous, then \( f(x) = \sin x \).
Answer: (B). This is the contrapositive of the given statement. In logic, the contrapositive of "if \( p \) then \( q \)" is "if not \( q \) then not \( p \)".
True or False: If \( f(x) \) is continuous everywhere, then \( |f(x)| \) is continuous everywhere.
Answer: True. This follows from the theorem on the composition of continuous functions: both \( f(x) \) and \( |x| \) are continuous everywhere.
True or False: If \( f(x) \) is continuous everywhere, then \( \sqrt{f(x)} \) is continuous everywhere.
Answer: False. \( \sqrt{f(x)} \) is only continuous where \( f(x) \ge 0 \).
True or False: If the composition \( f \circ g \) is not continuous at \( x = a \), then either \( g \) is not continuous at \( x = a \) or \( f \) is not continuous at \( g(a) \).
Answer: True. This is a consequence of the theorem on the composition of continuous functions.
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