GU4061: Intro Modern Analysis I
Final Practice Exam
Name:
UNI:
ˆ Each problem is worth 15 points, for a total of 120 points.
ˆ Unlike the midterm, this exam is double-sided—so be sure to solve the problems on both sides of
each page!
ˆ Answer the questions in the spaces provided. If you run out of room for an answer, continue on one of
the blank pages at the end of the exam packet. If you are doing this, make sure to write clearly on the
page with the original problem that I should check the later page for the rest of the solution. Otherwise,
I won’t know to check and you may not get credit for your work!
ˆ Remember to justify your answers fully. You must clearly state every single step in each of your proofs to
receive full credit. In writing your proofs, you may use any of the lemmas/theorems/propositions/corollaries
that we stated in class, or any result stated in a homework problem, but you may not assume any other
results.
ˆ Answer the questions as best you can; partial credit will be given for incomplete proofs.
ˆ Use of calculators, technology aids or study materials (textbooks, notes, etc.) is not permitted.
1. (3 points each) True or false; justify each answer with a proof or a counterexample.
(a) If X is a metric space, f : X → X is continuous, and U ⊂ X is connected, then f −1 (U ) is connected.
(b) If X and Y are compact metric spaces and f : X → Y is continuous, then f (X) is a closed subset of
Y.
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(c) If fn : [0, 1] →
R is defined as fn(x) = enx, then the sequence of functions {fn} is equicontinuous.
R
(d) If f : [0, 1] → is Riemann integrable, then the function F : [0, 1] →
is Riemann integrable.
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R given by F (x) = R0x f (t)dt
(e) If fn : [0, 1] →
R is defined as fn(x) = xn, then the sequence of functions {fn′ } is uniformly bounded.
2. (3 points each) True or false; justify each answer with a proof or a counterexample.
(a) If X is a metric space, then every Cauchy sequence in C(X) converges.
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(b) If X is a complete metric space, then X is compact.
(c) If f : [0, 1] → [0, 1] is a continuous bijection, then f −1 is continuous.
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(d) If f : [0, 1] →
[0, 1].
(e) If f :
R is monotonically increasing, then U (f, P ) − L(f, P ) < 1 for some partition P of
R → R is continuous, then f is differentiable at x for some x ∈ R.
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3. Let f : [0, 1] →
R be a bounded Riemann integrable function, and define F : [0, 1] → R as
Z
F (x) :=
x
f (t)dt.
0
Let γ : [0, 1] →
R2 be the function defined as γ(x) = (x, F (x)).
(a) (5 points) Show that γ is a curve.
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(b) (10 points) Show that γ is rectifiable.
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4. Let f :
R → R be a differentiable function.
(a) (5 points) Give an example (with justification) to show that f is not necessarily uniformly continuous.
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(b) (10 points) Suppose that there exists M > 0 such that |f ′ (x)| ≤ M for all x ∈
uniformly continuous.
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R. Show that f is
5. Suppose that f : [0, 1] →
R is continuous, and that
1
Z
f (x)xn dx = 0
0
for each nonnegative integer n.
(a) (3 points) Show that
Z
1
f (x)p(x)dx = 0
for every polynomial p : [0, 1] →
R.
0
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(b) (8 points) Use the result of part (a) to show that
Z
1
(f (x))2 dx = 0.
0
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(c) (4 points) Use the result of part (b) to show that f (x) = 0 for all x ∈ [0, 1].
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6. For each n ∈
N, let fn : (−1/2, 1/2) → R be the function fn(x) = xn/n.
(a) (7 points) Show that the series
∞
X
fn (x)
n=1
converges uniformly on (−1/2, 1/2) to a function f : (−1/2, 1/2) →
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R.
(b) (8 points) Show that the series
∞
X
fn′ (x)
n=1
′
converges uniformly on (−1/2, 1/2) to f .
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7. (a) (7 points) Show that, if f :
R → Q is continuous, then f is constant.
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(b) (8 points) Show that the map f :
Q → R given by
(
f (x) =
0
1
is continuous.
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x2 < 2
x2 > 2
8. Define the function f : [−1, 1] →
R as
(
x2
f (x) =
0
if x is rational
if x is irrational
(a) (7 points) Show that f is differentiable at 0 but not differentiable at any other point in (−1, 1).
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(b) (8 points) Is f Riemann integrable? Justify your answer with a proof.
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Use this page if you need extra room for one of your solutions. If you do use this page for one of your
solutions, remember to indicate this where the problem is stated, so I know to look here.
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Use this page if you need extra room for one of your solutions. If you do use this page for one of your
solutions, remember to indicate this where the problem is stated, so I know to look here.
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Use this page if you need extra room for one of your solutions. If you do use this page for one of your
solutions, remember to indicate this where the problem is stated, so I know to look here.
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Use this page if you need extra room for one of your solutions. If you do use this page for one of your
solutions, remember to indicate this where the problem is stated, so I know to look here.
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Use this page if you need extra room for one of your solutions. If you do use this page for one of your
solutions, remember to indicate this where the problem is stated, so I know to look here.
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