A Summary on Analysis 2
Chapter 7 Integration
Let 𝐺 ⊆ ℝ2 and 𝑃 be a partition of 𝐺. Then the under sum of 𝐺 , only if nonnegative, equals at most
the volume of the region under the graph. We define the integral of 𝑓 over 𝐺 as the supremum of
this set of numbers and denote the integral of 𝑓 over 𝐺 (the double integral) as:
∬ 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦
𝐺
Theorem 7.1 Integration by substitution for double integrals
Let 𝑔1 and 𝑔2 be continuously differentiable functions on a compact set 𝐺 ⊆ ℝ2 such, that the vector
valued function 𝑔 = (𝑔1 , 𝑔2 ) is an invertible function on 𝐺. Let 𝑓 be a continuous function on the
image 𝐺̃ = 𝑔(𝐺). Substitution 𝑢 = 𝑔1 (𝑥, 𝑦) and 𝑣 = 𝑔2 (𝑥, 𝑦) We have:
∬ 𝑓(𝑔(𝑥, 𝑦))|det 𝐷𝑔(𝑥, 𝑦)|𝑑𝑥𝑑𝑦 = ∬ 𝑓(𝑢, 𝑣)𝑑𝑢𝑑𝑣
𝐺 𝐺̃
𝐷1 𝑔1 (𝑥, 𝑦) 𝐷2 𝑔1 (𝑥, 𝑦)
det 𝐷𝑔(𝑥, 𝑦) = det [ ]
𝐷1 𝑔2 (𝑥, 𝑦) 𝐷2 𝑔2 (𝑥, 𝑦)
,Chapter 1 The 𝑛-Dimensional Euclidean Space
Definition
𝑥1
Let 𝑛 ∈ ℕ and 𝑥1 , … , 𝑥𝑛 an 𝑛-tuple real numbers. 𝒙 = [ ⋮ ]
𝑥𝑛
Is called the 𝒏-dimensional vector or a point in the 𝒏-dimensional space.
The set of all 𝑛-dimensional vectors 𝒙 is called the 𝒏-dimensional Euclidean space ℝ𝑛
The following hold in the Euclidean space:
- The 𝑛-dimensional zero vector is denoted by 𝟎. The 𝑛-dimensional vector for which 𝑒𝑖 = 1
with 𝑖 = 1, . . , 𝑛 is denoted by 𝒆.
- The set of all 𝑛-dimensional nonnegative vectors is denoted by ℝ𝑛+
A vector 𝒙 is nonnegative if 𝑥𝑖 ≥ 0 for every 𝑖 and is called a positive vector if 𝑥𝑖 > 0 for
every 𝑖. The set of all 𝑛-dimensional positive vectors is denoted by ℝ𝑛++
Definition (Revision of Linear Algebra)
𝑥1 + 𝑦1
𝒙+𝒚=[ ⋮ ] is called the sum vector of 𝒙 and 𝒚.
𝑥𝑛 + 𝑦𝑛
𝛼𝑥1
𝛼𝒙 = [ ⋮ ] is called the scalar product vector of 𝒙 and 𝛼
𝛼𝑥𝑛
⊤
𝒙 = [𝑥1 … 𝑥𝑛 ] is called the transposed vector of 𝒙
∥ 𝒙 ∥= √𝑥12 + ⋯ + 𝑥𝑛2 is called the Euclidean norm of 𝒙
Theorem 1.1 Inequality of Cauchy-Schwarz
Let 𝒙, 𝒚 ∈ ℝ𝑛 we have: |𝒙⊤ 𝒚| ≤∥ 𝒙 ∥∥ 𝒚 ∥
Proof
Define 𝑝(𝑡) = (𝑥1 − 𝑡𝑦1 )2 + ⋯ + (𝑥𝑛 − 𝑡𝑦𝑛 )2 = 𝐴𝑡 2 − 2𝐵 + 𝐶 where 𝐴 = ∥ 𝒚 ∥2 , 𝐶 = ∥ 𝒙 ∥2 and
𝐵 = |𝑥 ⊤ 𝑦|. Then 𝑝(𝑡) ≥ 0 (sum of squares) if 𝐷 ≤ 0 so (−2𝐵)2 − 4𝐴𝐶 = 𝐵2 − 𝐴𝐶 ≤ 0 ⇔ 𝐵2 ≤
𝐴𝐶 ⇔
|𝒙⊤ 𝒚| ≤∥ 𝒙 ∥∥ 𝒚 ∥
Theorem 1.2 Properties norm
Let 𝒙, 𝒚 ∈ ℝ𝑛 then we have:
- ∥ 𝒙 ∥≥ 0 and ∥ 𝒙 ∥= 0 iff. 𝒙 = 𝟎
- ∥ 𝛼𝒙 ∥= |𝛼| ∥ 𝒙 ∥
- ∥ 𝒙 + 𝒚 ∥≤∥ 𝒙 ∥ +∥ 𝒚 ∥ (Triangle inequality of the norm)
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 + 𝑏1
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 + 𝑏2
𝑓(𝒙) = 𝐴𝒙 + 𝒃 = [ ]
⋮
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 + 𝑏𝑚
,Here 𝑓: ℝ𝑛 → ℝ𝑚 is called an affine function, that is: the function is composed of a linear function +
a constant (graph is a straight line) .The vector (constant) is called the inhomogeneous part of 𝑓
𝑎11 𝑎12 … 𝑎1𝑛
𝑎 𝑎22 … 𝑎2𝑛
Let 𝐴 = [ 21 ] be the std matrix of 𝑓 on ℝ𝑛 defined by
⋮ ⋮
𝑎𝑛1 𝑎𝑛2 … 𝑎𝑛𝑛
𝑓(𝑥) = 𝑥 ⊤ 𝐴𝑥 = ∑𝑛𝑖=1 ∑𝑛𝑗=1 𝑎𝑖𝑗 𝑥𝑖 𝑥𝑗 then 𝑓 is called a quadratic function from ℝ𝑛 to ℝ
Definition 𝜺-neighborhood
Let 𝒂 ∈ ℝ𝑛 and 𝜀 > 0. The set 𝑈𝜀 (𝒂) = {𝒙 ∈ ℝ𝑛 : ∥ 𝒙 − 𝒂 ∥< 𝜀}
Is called the 𝜺-neighborhood of 𝑎 in ℝ𝑛
Definition
If ∃𝜀 > 0 such that 𝑈𝜀 (𝒂) ⊆ 𝑉, then 𝒂 is called an interior point of 𝑉.
If 𝒂 is not an interior point of 𝑉, then 𝒂 is called a boundary point of 𝑉.
Definition Open/ Closed set
A set 𝑉 ⊆ ℝ𝑛 is called an open set, if ∀𝑐 ∈ 𝑉∃𝜀 > 0 ∶ 𝑈𝜀 (𝒂) ⊆ 𝑉
A set 𝑉 ⊆ ℝ𝑛 is called a closed set, if the complement of 𝑉 in ℝ𝑛 (ℝ𝑛 \𝑉) is an open set.
Theorem 1.3 Intersection Open Sets
The intersection of two open subsets 𝑉 and 𝑊 of ℝ𝑛 is an open set
Definition Bounded
A set 𝑉 ⊆ ℝ𝑛 is called bounded if ∃𝑚 > 0 such that ∀𝒙 ∈ 𝑉 we have ||𝒙|| ≤ 𝑚
Definition Compact Set
A set 𝑉 ⊆ ℝ𝑛 is called compact, if 𝑉 is bounded an closed.
Definition
A sequence in ℕ → ℝ𝑛 is a function on the set of natural numbers
The set of all terms of the sequence is called the value set or the range of the sequence and is
denoted by: {𝑡𝑘 : 𝑘 ∈ ℕ}
Important to note is that the coordinates of the 𝑘-th term 𝑡𝑘 = (𝑡𝑘1 , 𝑡𝑘2 , … , 𝑡𝑘𝑛 ). So for every
𝑖 ∈ {1, … , 𝑛} the sequence of real numbers (𝑡𝑘𝑖 )∞
𝑘=1 is called the 𝑖-th component sequence or
∞
coordinate sequence of the sequence (𝑡𝑘 )𝑘=1
Theorem Limit of sequence + Definition Convergent Sequence
Let (𝑡𝑘 )∞ 𝑛 ∞
𝑘=1 be a sequence in ℝ . (𝑡𝑘 )𝑘=1 has limit 𝑙 if:
𝑙𝑖𝑚 ∥𝑡𝑘 − 𝑙∥∥ = 0
𝑘→∞
If a sequence has limit 𝑙, we say the sequence converges to 𝑙. 𝑙𝑖𝑚 𝑡𝑘 = 𝑙 or 𝑡𝑘 ⟶ 𝑙 if 𝑘 → ∞
𝑘→∞
A sequence that has a limit is called a convergent sequence and one that hasn’t is called a divergent
sequence.
,Important Note: ∥𝑡𝑘 − 𝑙∥∥ is the norm of 𝑡𝑘 − 𝑙
Theorem 1.4 Coordinate criterion for a convergent sequence
For every sequence (𝑡𝑘 )∞ 𝑛 𝑛
𝑘=1 in ℝ and 𝑙 ∈ ℝ we have:
𝑙𝑖𝑚 𝑡𝑘1 = 𝑙1
𝑘→∞
𝑙𝑖𝑚 𝑡𝑘2 = 𝑙2
𝑙𝑖𝑚 𝑡𝑘 = 𝑙 if and only if 𝑘→∞
𝑘→∞ ⋮ ⋮
{ 𝑙𝑖𝑚 𝑡𝑘𝑛 = 𝑙𝑛
𝑘→∞
Theorem 1.5 Theorem of Bolzano-Weierstrass for sequences in ℝ𝒏
Every bounded sequence in ℝ𝑛 has a convergent subsequence
Chapter 2 Limits of Functions and Continuity
Definition Accumulation Point
Let 𝑉 ⊆ ℝ𝑛 . A point 𝑐 ∈ ℝ𝑛 is called an accumulation point of 𝑉 if ∀𝜀 > 0 ∃𝑥 ∈ 𝑉 such that
0 <∥ 𝑥 − 𝑐 ∥< 𝜀
Interpretation: A point 𝑐 ∈ ℝ𝑛 is an accumulation point of 𝑉 if every 𝑈𝜀 (𝒄) contains a point 𝒙 such
that 𝑥 ≠ 𝑐 and 𝑥 ∈ 𝑉. However that does not mean 𝑐 ∈ 𝑉 so 𝑐 does not have to be an interior or
boundary point.
Definition Limit
Let 𝑓 be a function 𝑓: ℝ𝑛 → ℝ𝑚 and 𝑐 ∈ ℝ𝑛 an accumulation point of 𝐷𝑓 .
An 𝑚-dimensional point 𝑙 ∈ ℝ𝑚 is called the limit of 𝑓 in 𝑐, if for ∀𝜀 > 0 ∃𝛿 > 0 ∀𝑥 ∈ 𝐷𝑓 such that
0 <∥ 𝑥 − 𝑐 ∥< 𝛿, we have ∥ 𝑓(𝑥) − 𝑙 ∥< 𝜀
If 𝑙 is the limit of 𝑓 in 𝑐 then we say that 𝑓 tends to 𝑙 if 𝑥 tends to 𝑐: 𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 or 𝑓(𝑥) → 𝑙 if 𝑥 → 𝑐
𝑥→𝑐
Theorem 2.1 Coordinate Criterion for Limit of a Function
Let 𝑓 (𝐷𝑓 ⊆ ℝ𝑛 ) be a function 𝑓: ℝ𝑛 → ℝ𝑚 , 𝑐 ∈ ℝ𝑛 and 𝑙 ∈ ℝ𝑚 . We have:
𝑙𝑖𝑚 𝑓1 (𝑥) = 𝑙1
𝑥→𝑐
𝑙𝑖𝑚 𝑓2 (𝑥) = 𝑙2
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 if and only if 𝑥→𝑐
𝑥→𝑐 ⋮ ⋮
{𝑥→𝑐 𝑚 (𝑥)
𝑙𝑖𝑚 𝑓 = 𝑙𝑚
Proof
"⇒"
Let 𝑖 ∈ {1, … , 𝑚} and 𝜀 > 0. For 𝜀, ∃𝛿 > 0 such that ∀𝑥 ∈ 𝐷𝑓 with 0 <∥ 𝑥 − 𝑐 ∥< 𝛿 we have
∥ 𝑓(𝑥) − 𝑙 ∥< 𝜀. Then we have for 𝑥 ∈ 𝐷𝑓 with 0 <∥ 𝑥 − 𝑐 ∥< 𝛿: |𝑓𝑖 (𝑥) − 𝑙𝑖 | ≤∥ 𝑓(𝑥) − 𝑙 ∥< 𝜀
So 𝑙𝑖𝑚 𝑓𝑖 (𝑥) = 𝑙𝑖
𝑥→𝑐
, "⇐"
𝜀
Let 𝜀 > 0. For there exists for every 𝑖 ∈ {1, … , 𝑚} a 𝛿𝑖 > 0 such that ∀𝑥 ∈ 𝐷𝑓 with
√𝑚
𝜀
0 <∥ 𝑥 − 𝑐 ∥< 𝛿𝑖 we have |𝑓𝑖 (𝑥) − 𝑙𝑖 | < . Take 𝛿 = 𝑚𝑖𝑛{𝛿1 , … , 𝛿𝑚 } and let 𝑥 ∈ 𝐷𝑓 with
√𝑚
𝜀 2
0 <∥ 𝑥 − 𝑐 ∥< 𝛿 , then ∥ 𝑓(𝑥) − 𝑙 ∥< √𝑚 ( ) =𝜀
√𝑚
Theorem 2.2 Linking Limit Lemma
Let 𝑓: ℝ𝑛 → ℝ, 𝑐 ∈ ℝ𝑛 and 𝑙 ∈ ℝ. We have:
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 iff. for every sequence (𝑥𝑘 )∞
𝑘=1 in 𝐷𝑓 , such that 𝑥𝑘 ≠ 𝑐 for ∀𝑘 ∈ ℕ and 𝑙𝑖𝑚 𝑥𝑘 = 𝑐
𝑥→𝑐 𝑘→∞
we have that 𝑙𝑖𝑚 𝑓(𝑥𝑘 ) = 𝑙
𝑘→∞
Definition Continuity
Let 𝑓: ℝ𝑛 → ℝ𝑚 and 𝑐 ∈ D𝑓 where 𝑐 is an accumulation point of D𝑓 .
𝑓 is called continuous in 𝒄 if for ∀𝜀 > 0, ∃𝛿 > 0 ∀𝑥 ∈ 𝐷𝑓 with ∥ 𝑥 − 𝑐 ∥< 𝛿 we have:
∥ 𝑓(𝑥) − 𝑓(𝑐) ∥< 𝜀
A function 𝑓 is called continuous if 𝑓 is continuous in every point of 𝐷𝑓
Theorem 2.3 Coordinate Criterion for a Continuous Function
𝑓: ℝ𝑛 → ℝ𝑚 (vector valued) is continuous in a point iff. every coordinate function is continuous in
that point
Theorem 2.4 Linking Limit Lemma for Continuity of a Function
For a real valued function 𝑓 with 𝐷𝑓 ⊆ ℝ𝑛 we have:
𝑓 is continuous in 𝑐 iff. for every sequence (𝑥𝑘 )∞
𝑘=1 in 𝐷𝑓 , with 𝑙𝑖𝑚 𝑥𝑘 = 𝑐 , we have:
𝑘→∞
𝑙𝑖𝑚 𝑓(𝑥𝑘 ) = 𝑓(𝑐)
𝑘→∞
Theorem 2.5 Arithmetic Rules for Continuous Functions
For every pair of real valued functions 𝑓 and 𝑔 with 𝐷𝑓 = 𝐷𝑔 = 𝐷 ⊆ ℝ𝑛 and 𝑐 ∈ 𝐷 , we have:
If 𝑓 and 𝑔 are continuous in 𝑐, then
• 𝑓 + 𝑔 is continuous in 𝑐
• 𝑓 ∙ 𝑔 is continuous in 𝑐
𝑓
• 𝑔
is continuous in 𝑐
Theorem 2.6 Continuity of the Composite Function
Let 𝐷𝑓 ⊆ ℝ𝑛 and 𝐷𝑔 ⊆ ℝ𝑝 with 𝑔: ℝ𝑝 → ℝ𝑛 such that 𝑅𝑔 ⊆ 𝐷𝑓 and let 𝑐 ∈ 𝐷𝑔 , we have:
If 𝑔 is continuous in 𝑐 and 𝑓 is continuous in 𝑔(𝑐), then the composite real valued function 𝑓 ∘ 𝑔 is
continuous in 𝑐
Chapter 7 Integration
Let 𝐺 ⊆ ℝ2 and 𝑃 be a partition of 𝐺. Then the under sum of 𝐺 , only if nonnegative, equals at most
the volume of the region under the graph. We define the integral of 𝑓 over 𝐺 as the supremum of
this set of numbers and denote the integral of 𝑓 over 𝐺 (the double integral) as:
∬ 𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦
𝐺
Theorem 7.1 Integration by substitution for double integrals
Let 𝑔1 and 𝑔2 be continuously differentiable functions on a compact set 𝐺 ⊆ ℝ2 such, that the vector
valued function 𝑔 = (𝑔1 , 𝑔2 ) is an invertible function on 𝐺. Let 𝑓 be a continuous function on the
image 𝐺̃ = 𝑔(𝐺). Substitution 𝑢 = 𝑔1 (𝑥, 𝑦) and 𝑣 = 𝑔2 (𝑥, 𝑦) We have:
∬ 𝑓(𝑔(𝑥, 𝑦))|det 𝐷𝑔(𝑥, 𝑦)|𝑑𝑥𝑑𝑦 = ∬ 𝑓(𝑢, 𝑣)𝑑𝑢𝑑𝑣
𝐺 𝐺̃
𝐷1 𝑔1 (𝑥, 𝑦) 𝐷2 𝑔1 (𝑥, 𝑦)
det 𝐷𝑔(𝑥, 𝑦) = det [ ]
𝐷1 𝑔2 (𝑥, 𝑦) 𝐷2 𝑔2 (𝑥, 𝑦)
,Chapter 1 The 𝑛-Dimensional Euclidean Space
Definition
𝑥1
Let 𝑛 ∈ ℕ and 𝑥1 , … , 𝑥𝑛 an 𝑛-tuple real numbers. 𝒙 = [ ⋮ ]
𝑥𝑛
Is called the 𝒏-dimensional vector or a point in the 𝒏-dimensional space.
The set of all 𝑛-dimensional vectors 𝒙 is called the 𝒏-dimensional Euclidean space ℝ𝑛
The following hold in the Euclidean space:
- The 𝑛-dimensional zero vector is denoted by 𝟎. The 𝑛-dimensional vector for which 𝑒𝑖 = 1
with 𝑖 = 1, . . , 𝑛 is denoted by 𝒆.
- The set of all 𝑛-dimensional nonnegative vectors is denoted by ℝ𝑛+
A vector 𝒙 is nonnegative if 𝑥𝑖 ≥ 0 for every 𝑖 and is called a positive vector if 𝑥𝑖 > 0 for
every 𝑖. The set of all 𝑛-dimensional positive vectors is denoted by ℝ𝑛++
Definition (Revision of Linear Algebra)
𝑥1 + 𝑦1
𝒙+𝒚=[ ⋮ ] is called the sum vector of 𝒙 and 𝒚.
𝑥𝑛 + 𝑦𝑛
𝛼𝑥1
𝛼𝒙 = [ ⋮ ] is called the scalar product vector of 𝒙 and 𝛼
𝛼𝑥𝑛
⊤
𝒙 = [𝑥1 … 𝑥𝑛 ] is called the transposed vector of 𝒙
∥ 𝒙 ∥= √𝑥12 + ⋯ + 𝑥𝑛2 is called the Euclidean norm of 𝒙
Theorem 1.1 Inequality of Cauchy-Schwarz
Let 𝒙, 𝒚 ∈ ℝ𝑛 we have: |𝒙⊤ 𝒚| ≤∥ 𝒙 ∥∥ 𝒚 ∥
Proof
Define 𝑝(𝑡) = (𝑥1 − 𝑡𝑦1 )2 + ⋯ + (𝑥𝑛 − 𝑡𝑦𝑛 )2 = 𝐴𝑡 2 − 2𝐵 + 𝐶 where 𝐴 = ∥ 𝒚 ∥2 , 𝐶 = ∥ 𝒙 ∥2 and
𝐵 = |𝑥 ⊤ 𝑦|. Then 𝑝(𝑡) ≥ 0 (sum of squares) if 𝐷 ≤ 0 so (−2𝐵)2 − 4𝐴𝐶 = 𝐵2 − 𝐴𝐶 ≤ 0 ⇔ 𝐵2 ≤
𝐴𝐶 ⇔
|𝒙⊤ 𝒚| ≤∥ 𝒙 ∥∥ 𝒚 ∥
Theorem 1.2 Properties norm
Let 𝒙, 𝒚 ∈ ℝ𝑛 then we have:
- ∥ 𝒙 ∥≥ 0 and ∥ 𝒙 ∥= 0 iff. 𝒙 = 𝟎
- ∥ 𝛼𝒙 ∥= |𝛼| ∥ 𝒙 ∥
- ∥ 𝒙 + 𝒚 ∥≤∥ 𝒙 ∥ +∥ 𝒚 ∥ (Triangle inequality of the norm)
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 + 𝑏1
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 + 𝑏2
𝑓(𝒙) = 𝐴𝒙 + 𝒃 = [ ]
⋮
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 + 𝑏𝑚
,Here 𝑓: ℝ𝑛 → ℝ𝑚 is called an affine function, that is: the function is composed of a linear function +
a constant (graph is a straight line) .The vector (constant) is called the inhomogeneous part of 𝑓
𝑎11 𝑎12 … 𝑎1𝑛
𝑎 𝑎22 … 𝑎2𝑛
Let 𝐴 = [ 21 ] be the std matrix of 𝑓 on ℝ𝑛 defined by
⋮ ⋮
𝑎𝑛1 𝑎𝑛2 … 𝑎𝑛𝑛
𝑓(𝑥) = 𝑥 ⊤ 𝐴𝑥 = ∑𝑛𝑖=1 ∑𝑛𝑗=1 𝑎𝑖𝑗 𝑥𝑖 𝑥𝑗 then 𝑓 is called a quadratic function from ℝ𝑛 to ℝ
Definition 𝜺-neighborhood
Let 𝒂 ∈ ℝ𝑛 and 𝜀 > 0. The set 𝑈𝜀 (𝒂) = {𝒙 ∈ ℝ𝑛 : ∥ 𝒙 − 𝒂 ∥< 𝜀}
Is called the 𝜺-neighborhood of 𝑎 in ℝ𝑛
Definition
If ∃𝜀 > 0 such that 𝑈𝜀 (𝒂) ⊆ 𝑉, then 𝒂 is called an interior point of 𝑉.
If 𝒂 is not an interior point of 𝑉, then 𝒂 is called a boundary point of 𝑉.
Definition Open/ Closed set
A set 𝑉 ⊆ ℝ𝑛 is called an open set, if ∀𝑐 ∈ 𝑉∃𝜀 > 0 ∶ 𝑈𝜀 (𝒂) ⊆ 𝑉
A set 𝑉 ⊆ ℝ𝑛 is called a closed set, if the complement of 𝑉 in ℝ𝑛 (ℝ𝑛 \𝑉) is an open set.
Theorem 1.3 Intersection Open Sets
The intersection of two open subsets 𝑉 and 𝑊 of ℝ𝑛 is an open set
Definition Bounded
A set 𝑉 ⊆ ℝ𝑛 is called bounded if ∃𝑚 > 0 such that ∀𝒙 ∈ 𝑉 we have ||𝒙|| ≤ 𝑚
Definition Compact Set
A set 𝑉 ⊆ ℝ𝑛 is called compact, if 𝑉 is bounded an closed.
Definition
A sequence in ℕ → ℝ𝑛 is a function on the set of natural numbers
The set of all terms of the sequence is called the value set or the range of the sequence and is
denoted by: {𝑡𝑘 : 𝑘 ∈ ℕ}
Important to note is that the coordinates of the 𝑘-th term 𝑡𝑘 = (𝑡𝑘1 , 𝑡𝑘2 , … , 𝑡𝑘𝑛 ). So for every
𝑖 ∈ {1, … , 𝑛} the sequence of real numbers (𝑡𝑘𝑖 )∞
𝑘=1 is called the 𝑖-th component sequence or
∞
coordinate sequence of the sequence (𝑡𝑘 )𝑘=1
Theorem Limit of sequence + Definition Convergent Sequence
Let (𝑡𝑘 )∞ 𝑛 ∞
𝑘=1 be a sequence in ℝ . (𝑡𝑘 )𝑘=1 has limit 𝑙 if:
𝑙𝑖𝑚 ∥𝑡𝑘 − 𝑙∥∥ = 0
𝑘→∞
If a sequence has limit 𝑙, we say the sequence converges to 𝑙. 𝑙𝑖𝑚 𝑡𝑘 = 𝑙 or 𝑡𝑘 ⟶ 𝑙 if 𝑘 → ∞
𝑘→∞
A sequence that has a limit is called a convergent sequence and one that hasn’t is called a divergent
sequence.
,Important Note: ∥𝑡𝑘 − 𝑙∥∥ is the norm of 𝑡𝑘 − 𝑙
Theorem 1.4 Coordinate criterion for a convergent sequence
For every sequence (𝑡𝑘 )∞ 𝑛 𝑛
𝑘=1 in ℝ and 𝑙 ∈ ℝ we have:
𝑙𝑖𝑚 𝑡𝑘1 = 𝑙1
𝑘→∞
𝑙𝑖𝑚 𝑡𝑘2 = 𝑙2
𝑙𝑖𝑚 𝑡𝑘 = 𝑙 if and only if 𝑘→∞
𝑘→∞ ⋮ ⋮
{ 𝑙𝑖𝑚 𝑡𝑘𝑛 = 𝑙𝑛
𝑘→∞
Theorem 1.5 Theorem of Bolzano-Weierstrass for sequences in ℝ𝒏
Every bounded sequence in ℝ𝑛 has a convergent subsequence
Chapter 2 Limits of Functions and Continuity
Definition Accumulation Point
Let 𝑉 ⊆ ℝ𝑛 . A point 𝑐 ∈ ℝ𝑛 is called an accumulation point of 𝑉 if ∀𝜀 > 0 ∃𝑥 ∈ 𝑉 such that
0 <∥ 𝑥 − 𝑐 ∥< 𝜀
Interpretation: A point 𝑐 ∈ ℝ𝑛 is an accumulation point of 𝑉 if every 𝑈𝜀 (𝒄) contains a point 𝒙 such
that 𝑥 ≠ 𝑐 and 𝑥 ∈ 𝑉. However that does not mean 𝑐 ∈ 𝑉 so 𝑐 does not have to be an interior or
boundary point.
Definition Limit
Let 𝑓 be a function 𝑓: ℝ𝑛 → ℝ𝑚 and 𝑐 ∈ ℝ𝑛 an accumulation point of 𝐷𝑓 .
An 𝑚-dimensional point 𝑙 ∈ ℝ𝑚 is called the limit of 𝑓 in 𝑐, if for ∀𝜀 > 0 ∃𝛿 > 0 ∀𝑥 ∈ 𝐷𝑓 such that
0 <∥ 𝑥 − 𝑐 ∥< 𝛿, we have ∥ 𝑓(𝑥) − 𝑙 ∥< 𝜀
If 𝑙 is the limit of 𝑓 in 𝑐 then we say that 𝑓 tends to 𝑙 if 𝑥 tends to 𝑐: 𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 or 𝑓(𝑥) → 𝑙 if 𝑥 → 𝑐
𝑥→𝑐
Theorem 2.1 Coordinate Criterion for Limit of a Function
Let 𝑓 (𝐷𝑓 ⊆ ℝ𝑛 ) be a function 𝑓: ℝ𝑛 → ℝ𝑚 , 𝑐 ∈ ℝ𝑛 and 𝑙 ∈ ℝ𝑚 . We have:
𝑙𝑖𝑚 𝑓1 (𝑥) = 𝑙1
𝑥→𝑐
𝑙𝑖𝑚 𝑓2 (𝑥) = 𝑙2
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 if and only if 𝑥→𝑐
𝑥→𝑐 ⋮ ⋮
{𝑥→𝑐 𝑚 (𝑥)
𝑙𝑖𝑚 𝑓 = 𝑙𝑚
Proof
"⇒"
Let 𝑖 ∈ {1, … , 𝑚} and 𝜀 > 0. For 𝜀, ∃𝛿 > 0 such that ∀𝑥 ∈ 𝐷𝑓 with 0 <∥ 𝑥 − 𝑐 ∥< 𝛿 we have
∥ 𝑓(𝑥) − 𝑙 ∥< 𝜀. Then we have for 𝑥 ∈ 𝐷𝑓 with 0 <∥ 𝑥 − 𝑐 ∥< 𝛿: |𝑓𝑖 (𝑥) − 𝑙𝑖 | ≤∥ 𝑓(𝑥) − 𝑙 ∥< 𝜀
So 𝑙𝑖𝑚 𝑓𝑖 (𝑥) = 𝑙𝑖
𝑥→𝑐
, "⇐"
𝜀
Let 𝜀 > 0. For there exists for every 𝑖 ∈ {1, … , 𝑚} a 𝛿𝑖 > 0 such that ∀𝑥 ∈ 𝐷𝑓 with
√𝑚
𝜀
0 <∥ 𝑥 − 𝑐 ∥< 𝛿𝑖 we have |𝑓𝑖 (𝑥) − 𝑙𝑖 | < . Take 𝛿 = 𝑚𝑖𝑛{𝛿1 , … , 𝛿𝑚 } and let 𝑥 ∈ 𝐷𝑓 with
√𝑚
𝜀 2
0 <∥ 𝑥 − 𝑐 ∥< 𝛿 , then ∥ 𝑓(𝑥) − 𝑙 ∥< √𝑚 ( ) =𝜀
√𝑚
Theorem 2.2 Linking Limit Lemma
Let 𝑓: ℝ𝑛 → ℝ, 𝑐 ∈ ℝ𝑛 and 𝑙 ∈ ℝ. We have:
𝑙𝑖𝑚 𝑓(𝑥) = 𝑙 iff. for every sequence (𝑥𝑘 )∞
𝑘=1 in 𝐷𝑓 , such that 𝑥𝑘 ≠ 𝑐 for ∀𝑘 ∈ ℕ and 𝑙𝑖𝑚 𝑥𝑘 = 𝑐
𝑥→𝑐 𝑘→∞
we have that 𝑙𝑖𝑚 𝑓(𝑥𝑘 ) = 𝑙
𝑘→∞
Definition Continuity
Let 𝑓: ℝ𝑛 → ℝ𝑚 and 𝑐 ∈ D𝑓 where 𝑐 is an accumulation point of D𝑓 .
𝑓 is called continuous in 𝒄 if for ∀𝜀 > 0, ∃𝛿 > 0 ∀𝑥 ∈ 𝐷𝑓 with ∥ 𝑥 − 𝑐 ∥< 𝛿 we have:
∥ 𝑓(𝑥) − 𝑓(𝑐) ∥< 𝜀
A function 𝑓 is called continuous if 𝑓 is continuous in every point of 𝐷𝑓
Theorem 2.3 Coordinate Criterion for a Continuous Function
𝑓: ℝ𝑛 → ℝ𝑚 (vector valued) is continuous in a point iff. every coordinate function is continuous in
that point
Theorem 2.4 Linking Limit Lemma for Continuity of a Function
For a real valued function 𝑓 with 𝐷𝑓 ⊆ ℝ𝑛 we have:
𝑓 is continuous in 𝑐 iff. for every sequence (𝑥𝑘 )∞
𝑘=1 in 𝐷𝑓 , with 𝑙𝑖𝑚 𝑥𝑘 = 𝑐 , we have:
𝑘→∞
𝑙𝑖𝑚 𝑓(𝑥𝑘 ) = 𝑓(𝑐)
𝑘→∞
Theorem 2.5 Arithmetic Rules for Continuous Functions
For every pair of real valued functions 𝑓 and 𝑔 with 𝐷𝑓 = 𝐷𝑔 = 𝐷 ⊆ ℝ𝑛 and 𝑐 ∈ 𝐷 , we have:
If 𝑓 and 𝑔 are continuous in 𝑐, then
• 𝑓 + 𝑔 is continuous in 𝑐
• 𝑓 ∙ 𝑔 is continuous in 𝑐
𝑓
• 𝑔
is continuous in 𝑐
Theorem 2.6 Continuity of the Composite Function
Let 𝐷𝑓 ⊆ ℝ𝑛 and 𝐷𝑔 ⊆ ℝ𝑝 with 𝑔: ℝ𝑝 → ℝ𝑛 such that 𝑅𝑔 ⊆ 𝐷𝑓 and let 𝑐 ∈ 𝐷𝑔 , we have:
If 𝑔 is continuous in 𝑐 and 𝑓 is continuous in 𝑔(𝑐), then the composite real valued function 𝑓 ∘ 𝑔 is
continuous in 𝑐
