•
Limit
A
⊂
R
,
f
:
A
→
R
,
c
∈
R
. The limit of f at c exists provided
1.
∃
a,b
∈
R
,a < c < b,
3
[(
a,c
)
∪
(
c,b
)]
⊂
A
2.
∃
L
∈
R
3 ∀
± >
0,
∃
δ >
0
3
if 0
<

x

c

< δ
, then

f
(
x
)

L

< ±
.
– limit at inﬁnity:
A
⊂
R
,
f
:
A
→
R
, then the limit as x goes to inﬁnity of
f
exists provided
1.
∃
a
∈
R
3
(
a,
∞
)
⊂
A
.
2.
∃
L
∈
R
3 ∀
± >
0
∃
x
o
∈
R
3
if
x > x
o
then

f
(
x
)

L

< ±
.
– limit in higher dimensions:
D
⊂
R
n
,
f
:
D
→
R
m
, and
p
o
∈
R
n
. The
limit of
f
at
p
o
exists provided that both
1.
p
o
is a cluster point of
D
.
2.
∃
L
∈
R
m
3 ∀
± >
0
∃
δ >
0
3
if
p
∈
D
and 0
<

p

p
o

< δ
then

f
(
p
)

L

< ±
.
– Thm:
p
o
∈
D
⊂
R
n
and
f
:
D
→
R
m
. Then
1. If
p
o
is an isolated point of
D
then
f
is continuous at
p
o
.
2. If
p
o
is a cluster point of
D
then
f
is continuous at
p
o
iﬀ lim
p
→
po
f
(
p
) =
f
(
p
o
).
•
Derivative
A
⊂
R
,
f
:
A
→
R
,
z
∈
A
.
f
is diﬀerentiable at
z
provided that
1.
z
∈
A
.
2.
∃
a,b
3
a <
0
< b
, and if
a < h <
0 or 0
< h < b
then
z
+
h
∈
A
.
3.
∃
L
3 ∀
± >
0
∃
δ >
0
3
if 0
<

h

< δ
then

f
(
z
+
h
)

f
(
z
)
h

L

< ±
.
Basically, lim
h
→
0
f
(
z
+
h
)

f
(
z
)
h
exists and equals
L
.
– Derivative in higher dimensions:
S
⊂
R
n
,
f
:
S
→
R
, and
p
o
is inte
rior point of
S
. Then
f
is diﬀerentiable at
p
o
provided
∃
V
∈
R
n
such that
lim
Δ
p
→
¯
0
f
(
po
+Δ
p
)

f
(
po
)

V
·
Δ
p

Δ
p

= 0. Note: if such a
V
exists, it is equal to
the gradient (
f
1
,f
2
,...,f
n
). Denoted
Df
(
p
o
).
– How to prove/disprove diﬀerentiability:
All you have to do is prove
that the gradient (vector of partial derivatives) is the derivative or that it is
not the derivative. If the partial derivatives don’t exist, then obviously the
derivative doesn’t exist either.
– Thm:
S
⊂
R
n
is open,
p
o
∈
S
and
f
:
S
→
R
is
C
1
. Then the derivative
Df
(
p
o
) exists.
– Thm:
If
S
⊂
R
n
,
f
:
S
→
R
and
p
o
is an interior point of
S
. If
f
is
diﬀerentiable at
p
o
, then
f
is continuous at
p
o
.
– Thm:
If
S
⊂
R
n
and
f
:
S
→
R
. If
f
is diﬀerentiable at
p
o
then
∀
β
∈
R
n
with

β

= 1,
D
B
f
(
p
o
) exists and equals
Df
(
p
o
)
·
β
.
– Derivative in highest dimensions:
S
⊂
R
n
,
f
:
S
→
R
m
, and
p
o
is
interior point of
S
. Then
f
is diﬀerentiable at
p
o
provided
∃
m
×
n
matrix
B
such that lim
Δ
p
→
¯
0
f
(
po
+Δ
p
)

f
(
po
)

B
Δ
p

Δ
p

=
¯
0
∈
R
m
.
B
is the Jacobian,
with partial derivatives of
f
1
in ﬁrst row, of
f
2
in second row, etc.
– Directional derivative