Problems graded: 29.3 (15 points), 29.13 (10 points), 32.2 (10 points); the
other problems were graded based on completion and worth five points each
(except that I counted 28.8, which was longer, as two problems).
To convert
from 65 points to 100, scores were multiplied by 1.5; then we gave everyone 2.5
points, rounding all fractions up at the end.
Final homework grade: Students have received nine scores, each out of 100.
As per the syllabus, we dropped the two lowest scores for each students and
converted the remaining sum to a percentage out of 100, rounding any fractional
part up (623/700 = 89 percent but 624/700 = 89.14 percent gets bumped up
to 90 percent).
28.8. (Note: I did this question in section on November 18.)
(a) Suppose
ǫ >
0. If

x
−
0

<
√
ǫ
then
f
(
x
) is equal to either
x
2
∈
[0
, ǫ
) or
0; in any event,

f
(
x
)
−
f
(0)

< ǫ
implying continuity at zero.
(b)
x
rational nonzero:
Let
ǫ
=
x
2
and suppose
δ >
0.
There exists an
irrational number
q
in (
x
−
δ, x
+
δ
) (see Exercise 4.12 from HW 2). However,
while

x
−
q

< δ
,

f
(
x
)
−
f
(
q
)

=
ǫ
;
δ
can be made arbitrarily small so
f
is NOT
continuous at
x
.
x
irrational: Let
ǫ
=
.
1
x
2
and suppose
δ >
0 (we may further suppose that
δ < .
5

x

). There exists a rational number
q
in (
x
−
δ, x
+
δ
); by the triangle
inequality,

q

> .
5

x

so
f
(
q
) =
.
25
x
2
. Therefore,

f
(
x
)
−
f
(
q
)

=
.
25
x
2
> ǫ
; as
δ
can be made arbitrarily small,
f
is not continuous at
x
.
As all nonzero points are either nonzero rational or irrational,
f
is discon
tinuous for all nonzero
x
.
(c) If
x
= 0 and
h
negationslash
= 0,
f
(
x
+
h
)
−
f
(
x
)
h
=
f
(
h
)
h
, which is equal to
h
if
h
is rational
and 0 otherwise. In any event, the expression clearly goes to zero as
h
goes to
zero (as it is bounded above in magnitude by

h

) so
f
is differentiable at
x
= 0
with derivative 0.
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 Spring '09
 WEISBART
 Metric space, Rational number, Irrational number, tk

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