practice problems for real analysis
TRANSCRIPT
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8/13/2019 Practice Problems for Real Analysis
1/12
Q Q R2
Q Q R2
A, B
A
B= Q Q
QQ QQ
E= R2
A1= (, ) (, ) B1= (, ) (, ) A1 B1
R R2 A1 B1
A= (Q Q)
A1 B= (Q Q)
B1 A
B Q Q A1 B1
(, ) (, ) (, ) (, )
R2 Q Q
QQ =A
B
Q Q R2
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8/13/2019 Practice Problems for Real Analysis
2/12
C={(x, y) : x2 +y2 = 1} R2
f p= [0, 2] R C={(x, y) : x2 + y2 = 1} R2
f(p) = (cos(p),sin(p))
R [0, 2]
f f
f(p) =cos(p) f(p) =sin(p)
f(p) =cos(p) p [0, 2] >0
>0 |f(p) f(p0)|< p [0, 2] |p p0|<
p0 [0, 2] = 2
cos(x) cos(y) =2 sin
x+y
2
sin
x y
2
|cos(p) cos(p0)|=
2 sin
p+p02
sin
p p0
2
sin(x) 1 sin
p+p0
2
1
2
sin
p p02
2
sin pp02
< |p p0|< pp02 < |sin(x)| |x|
2
sin
p p02
2p p02
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8/13/2019 Practice Problems for Real Analysis
3/12
f(p) = sin(p) p [0, 2] > 0
>0 |f(p) f(p0)|< p [0, 2] |p p0|<
p0 [0, 2] = 2
sin(x) sin(y) = 2 cos
x+y
2
sin
x y
2
|sin(p) sin(p0)|=
2 cos
p+p02
sin
p p0
2
cos(x) 1 cos
p+p0
2
1
2sin
p p02
2
sin pp02
< |p p0|<
pp02
< |sin(x)| |x|
2
sin
p p02
2p p02
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8/13/2019 Practice Problems for Real Analysis
4/12
(E, d) f, g : E R limpp0f(p) = q0
limpp0g(p) = r0 limpp0(f g)(p) = q0r0
limpp0f(p) = q0 > 0 1 > 0 |f(p) q0| <
0< |p p0|< 1 p E
limpp0g(p) =r0 > 0 2 > 0 |g(p) r0| <
0< |p p0|< 2 p E
>0 >0 |(f g)(p) q0r0|< 0< |p p0|<
2 + (|qo| + |r0|) = = = min{1, 2}
2 + (|qo| + |r0|)
0 f(p)r0 f(p)r0
|(f g)(p) q0r0|= |f(p)g(p) q0r0|= |f(p) (g(p) r0) +r0(f(p) q0)|
|x y| |x| |y|
|f(p)| |g(p) r0| + |r0| |f(p) q0|
|f(p) q0|< = |g(p) r0|<
=
(|f(p)| + |r0|)
0 |q0| | q0|
(|f(p) q0+q0| + |r0|)
(|f(p) q0| + |q0| + |r0|)
|f(p) q0|< =
(+ |q0| + |r0|) = 2 + (|qo| + |r0|)
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8/13/2019 Practice Problems for Real Analysis
5/12
2 + (|qo| + |r0|)
= 0 0 e=
0< 0 = min{1, 2} |(f g)(p) q0r0| < 0 0 1 > 0 |f(p) q0| <
0< |p p0|< 1 p E
limpp0h(p) = r0 > 0 2 > 0 |h(p) q0| <
0< |p p0|< 2 p E
f(p) g(p) h(p)
|p p0|< r
>0 >0 |g(p) q0|< 0< |p p0|< p E
|f(p) q0| < |h(p) q0| < q0 < f(p) < q0 +
q0 < h(p)< q0+
p E = min{1, 2, r} q0 < f(p) g(p) h(p)< q0+
0< |p p0|<
q0 < g(p)< q0+ |g(p) q0|<
>0 >0 |g(p) q0|< p E
0< |p p0|< = min{1, 2, r}
limpp0g(p) =q0.
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8/13/2019 Practice Problems for Real Analysis
7/12
f : R R 0
f(x) =
x if x Q
0 if x / Q 0
f 0
f x >0 >0
f(B(x)) B((f(x))
x = 0 f(x) = 0 B(0) = (, ). = e f(B(0)) B(0)
=
0
0
f R
a= 0 Q f(a) =a
>0 > 0 xR |x a| < f(B(a))
B((f(a)) f(B(a)) B(a)
= |a|2
B(a) 0
B(a) R
B(a) 0
B(a)
= |a|2
>0 x R |xa|< f(B(a)) B(a)
a= 0 / Q f(a) = 0
>0 >0 x R |x a|< f(B(a))
B((f(a)) f(B(a)) B(0)
= |a|2
B(0) a
B(a) R
B(a) a
B(0)
= |a|2
>0 x R |x a|< f(B(a))
B(0)
f 0 x R
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8/13/2019 Practice Problems for Real Analysis
8/12
f : R R |f|
f(x) =
1 if x Q
1 if x / Q |f(x)| = 1
x R
f
a Q
f(a) = 1
>0 > 0 xR |x a| < f(B(a))
B((f(a)) f(B(a)) B(1)
= 12
B(1) 1
B(a) R
B(a) 1
B(1)
= 12
>0 x R |x a|< f(B(a)) B(1)
a / Q f(a) =1
>0 >0 x R |x a|< f(B(a))
B((f(a)) f(B(a)) B(1)
= 12
B(1) 1
B(a) R
B(a) 1
B(1)
= 12
> 0 xR |x a|< f(B(a))
B(1)
f
|f(x)|= 1 x R
> 0 = x R |x a| < f(B(a)) B(1)
B(a) 1
a R |f|
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8/13/2019 Practice Problems for Real Analysis
9/12
f : Z R
p0 Z f(p0) =q0 R
>0 >0 |f(p) q0|< p Z |p p0|<
f(B(p0)) B(q0)
= 12
Z
B(p0) =p0 B(p0)
p0 p= p0
|f(p) f(p0)|= |f(p0) f(p0)|= 0< >0
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8/13/2019 Practice Problems for Real Analysis
10/12
d(x, y) =|x y| R d
R R = R2 d: R2 R
p0= (x0, y0) R2 d(p0) =|x0 y0| R
> 0 > 0 |d(p) d(p0)| < p = (x, y) R2
d(p, p0)<
d(p, p0)<
(x x0)2 + (y y0)2 0 >0 |f(p) q0|< |p p0|< p R
= f(x) = 0 x R
|f(p) q0|= |0 0|= 0<
>0
f
f U R
{0} R
{0} R\{0}
a
a
R\{0}
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8/13/2019 Practice Problems for Real Analysis
12/12
f : R R S
f1(S)
f(x) =x2 S= {1}
S
{1}.
f
x0 R f(x0) =x20 R
> 0 >0 |f(x) f(x0)|< |x x0|< x R
= |x+x0|
f
|f(x) f(x0)|= |x2 x20|= |(x x0)(x+x0)|
|x x0| |y y0|
|x x0|<
< |x+x0|=
|x+x0||x+x0|=
f R
f1(S) f1(S) ={1, 1}= C
C=A
B A B
C
C C
R
A= C
(, 0) B = C
(0, ) (, 0) (0, )
R A B C
A = 1 B = 1 C
C
f1(S)