Math 414: Analysis I
Exam 3 (80 points)
Spring 2014
Name:
*No notes or electronic devices. You must show all your work to receive full credit.
When justifying your answers, use only those techniques that we learned in class. Note
that there are 90 possible points on the test.
1. For each of the following statements, clearly circle the response that best reflects
the accuracy of the statement. If the statement is always true, circle ALWAYS.
If the statement is never true, circle NEVER. If the statement is sometimes true,
circle SOMETIMES. No justification for your answer is required. (2 points each, 10
points total)
(a) If f : [a, b] → R is a continuous function on [a, b], then f is Lipschitz continuous.
i. Always.
ii. Sometimes.
iii. Never.
(b) Suppose that f : [a, b] → R is a continuous function on [a, b] and f (x) > 0 for
all x ∈ [a, b]. Furthermore, suppose that f is a decreasing function on [a, b].
That is, if x < y, then f (x) > f (y) for all x, y ∈ [a, b]. Then
inf f (x) = 0.
(1)
x∈[a,b]
i. Always.
ii. Sometimes.
iii. Never.
(c) If f : S → R has the property that for any xn ∈ S such that xn → x, then
f (xn ) → f (x) for any x ∈ R, then f is uniformly continuous on S.
i. Always.
ii. Sometimes.
iii. Never.
(d) Suppose that f : [a, b] → R is a continuous function on [a, b]. Then there is a
c and d in R such that
f ([a, b]) = {f (x) : x ∈ [a, b]} = (c, d) = {y : c < y < d}
(2)
i. Always.
ii. Sometimes.
iii. Never.
(e) Let f : S → R be a function and c ∈ S. If c is not a cluster point for S, then
f is continuous at c.
i. Always.
ii. Sometimes.
iii. Never.
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Math 414: Analysis I
Exam 3 (80 points)
Spring 2014
|x|
. Using the
1 + |x|
− δ definition, prove that f is uniformly continuous on R.
2. (a) (10) Consider the function f : R → R defined by f (x) =
(b) (10) Prove that f : R → R defined by f (x) = x2 is not uniformly continuous
on R. (Hint: Consider one of your sequences as (xn ) = n + n1 )
3. (10) Suppose that f : [0, ∞) → R is a continuous function on [0, ∞) such that
|f (x)| ≤ x2 for all x ≥ 0. Prove that f is differentiable at x = 0.
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Math 414: Analysis I
Exam 3 (80 points)
Spring 2014
4. (15) Let f : [a, b] → R, and g : [a, b] → R be continuous functions. Define the set
E = {x ∈ [a, b] : f (x) = g(x)} .
Prove that if (xn ) is any sequence in E and xn → x0 , then x0 ∈ E.
5. (15) Suppose that f is continuous on [0, 1] and that f (0) < 0 and f (1) > 1. Prove
that there is at least one point c between 0 and 1 such that f (c) = c2 .
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Math 414: Analysis I
Exam 3 (80 points)
Spring 2014
6. (20) Suppose that f : I → R is a bounded function (i.e. there is an M > 0 such that
|f (x)| ≤ M for all x ∈ I), and suppose that g : I → R is a function differentiable
at c ∈ I and g(c) = g 0 (c) = 0. Prove that h(x) = f (x)g(x) is differentiable at c and
that h0 (c) = 0.
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