we can write the probability that we do not get a 6 as p (not 6) find the probabilty and show how yo
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University of California Los Angeles Isolated Point Gradescope Questions
1. Let A ⊆ R. We say that an element c ∈ A is isolated if there is an ε > 0 such that
A ∩ (c − ε, c + ε) = ...
University of California Los Angeles Isolated Point Gradescope Questions
1. Let A ⊆ R. We say that an element c ∈ A is isolated if there is an ε > 0 such that
A ∩ (c − ε, c + ε) = {c}.
(a) Show that c ∈ A is an isolated point if and only if it is not a limit point of A.
(b) Show that every function f : A → R is continuous at each isolated c ∈ A.
(c) Show that every function f : Z → R is continuous on its domain Z. 2. For each of the following functions f : A → R, find f(A) and hence decide whether f
(equivalently its range) has an upper bound, a lower bound, a maximum or a minimum.
(a) f(x) = x
3
, A = (−3, 2).
(b) f(x) = x
2
, A = (−3, 2)
(c) f(x) = (
x if x ∈ Q
0 if x /∈ Q
, A = [0, a] where a > 0. 3. Assume f : R → R is continuous on R and let K = {x : f(x) = 0}. Show that K is a
closed set. 4. Give an example of each of the following, or state that such a request is impossible. For
any that are impossible, supply a short explanation for why this is the case.
(a) A continuous function f : (0, 1) → R and a Cauchy sequence (xn) in (0, 1) such
that (f(xn)) is not a Cauchy sequence.
(b) A continuous function f : [0, 1] → R and a Cauchy sequence (xn) in [0, 1] such that
(f(xn)) is not a Cauchy sequence.
(c) A continuous function f : [0, 1] → R which has a maximum but no minimum.
(d) A continuous bounded function f : (0, 1) → R that attains a maximum value but
not a minimum value. 5. (a) Let f be a continuous real-valued function with domain (a, b). Show that if f(x) = 0
for each rational number x in (a, b), then f(x) = 0 for all x ∈ (a, b).
(b) If f and g are continuous real-valued functions with domain (a, b) and f(x) = g(x)
for all rational x ∈ (a, b), must f and g be the same function?
George Mason University Statistical Test Exam Questions
Solve each equation to the best of your ability. The attachment correlate with the some equations.
George Mason University Statistical Test Exam Questions
Solve each equation to the best of your ability. The attachment correlate with the some equations.
2 pages
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In 2010, the population of the United States is about 281 million people. The na
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Activity 3 Compound Interest Student Template
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University of California Los Angeles Isolated Point Gradescope Questions
1. Let A ⊆ R. We say that an element c ∈ A is isolated if there is an ε > 0 such that
A ∩ (c − ε, c + ε) = ...
University of California Los Angeles Isolated Point Gradescope Questions
1. Let A ⊆ R. We say that an element c ∈ A is isolated if there is an ε > 0 such that
A ∩ (c − ε, c + ε) = {c}.
(a) Show that c ∈ A is an isolated point if and only if it is not a limit point of A.
(b) Show that every function f : A → R is continuous at each isolated c ∈ A.
(c) Show that every function f : Z → R is continuous on its domain Z. 2. For each of the following functions f : A → R, find f(A) and hence decide whether f
(equivalently its range) has an upper bound, a lower bound, a maximum or a minimum.
(a) f(x) = x
3
, A = (−3, 2).
(b) f(x) = x
2
, A = (−3, 2)
(c) f(x) = (
x if x ∈ Q
0 if x /∈ Q
, A = [0, a] where a > 0. 3. Assume f : R → R is continuous on R and let K = {x : f(x) = 0}. Show that K is a
closed set. 4. Give an example of each of the following, or state that such a request is impossible. For
any that are impossible, supply a short explanation for why this is the case.
(a) A continuous function f : (0, 1) → R and a Cauchy sequence (xn) in (0, 1) such
that (f(xn)) is not a Cauchy sequence.
(b) A continuous function f : [0, 1] → R and a Cauchy sequence (xn) in [0, 1] such that
(f(xn)) is not a Cauchy sequence.
(c) A continuous function f : [0, 1] → R which has a maximum but no minimum.
(d) A continuous bounded function f : (0, 1) → R that attains a maximum value but
not a minimum value. 5. (a) Let f be a continuous real-valued function with domain (a, b). Show that if f(x) = 0
for each rational number x in (a, b), then f(x) = 0 for all x ∈ (a, b).
(b) If f and g are continuous real-valued functions with domain (a, b) and f(x) = g(x)
for all rational x ∈ (a, b), must f and g be the same function?
George Mason University Statistical Test Exam Questions
Solve each equation to the best of your ability. The attachment correlate with the some equations.
George Mason University Statistical Test Exam Questions
Solve each equation to the best of your ability. The attachment correlate with the some equations.
2 pages
Cmath 980
Under the assumption that the rate is constant, the relationship becomes the formula: distance = rate x 1. Suppose the dis ...
Cmath 980
Under the assumption that the rate is constant, the relationship becomes the formula: distance = rate x 1. Suppose the distance in miles traveled from ...
In 2010, the population of the United States is about 281 million people. The na
In 2010, the population of the United States is about 281 million people. The national debt is 13.6 trillion dollars ...
In 2010, the population of the United States is about 281 million people. The na
In 2010, the population of the United States is about 281 million people. The national debt is 13.6 trillion dollars (one trillion is 1012). How much would each person have to contribute to pay off the debt?
3 pages
Activity 3 Compound Interest Student Template
Reflection 1: What would be another way to calculate total interest rather than just adding up the individual interest Get ...
Activity 3 Compound Interest Student Template
Reflection 1: What would be another way to calculate total interest rather than just adding up the individual interest Getting the total amount at the ...
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