Continuities and Discontinuities of Real Valued Function

Continuity- A function is called continuous at x=a, if

                        (i)  f(x) is defined at x = a, i.e. f(x) = f(a) at x=a.

                       (ii) left and right hand limits exists at x=a.

                      (iii) f(a+) = f(a-) = f(a).

Discontinuity- A function f(x) which is not continuous at x=a, is called discontinuous at x=a.

Continuous at x₀- Let E ⊂ ℝ, f : E → ℝ, if x₀ ∈ E, then f is continuous at x₀ iff for each ε > 0 there exist δ > 0 : if
|x - x₀| < δ, x ∈ E, ⇒ |f(x) - f(x₀)| < ε.

If f is continuous at x for each x ∈ E, then f is continuous (on E).

Uniformly Continuity- A function f : D → ℝ is uniformly continuous on E ⊂ D iff for every ε > 0, there is δ > 0 : if 

x, y ∈ E with |x - y| < δ, then |f(x) - f(y)| < ε. If f  is uniformly continuous on D, f is uniformly continuous.

Closed set- A set E ⊂ ℝ is closed iff every accumulation point of E belongs to E.

Open set- A set A ⊂ ℝ is open iff for each x ∈ A, there is a neighbourhood N_x of x such that N_x ⊂ A.

Compact set- A set E is compact iff for every family 

{G_α}_α∈A of open sets E ⊂ ∪ G_α , α ∈ A, there is a finite set 

{α₁, . . . , α_n} ⊂ A : ⊂ ∪G_αi , i = 1, 2, ... , n.

Right  continuous-  If E ⊂ ℝ, f : E → ℝ, and x₀ ∈ E, the function f is right continuous at x₀ iff for each ε > 0, there is δ > 0 : x₀ ≤ x ≤ x + δ, x ∈ E ⇒ |f(x) - f(x₀)| < ε.

Left  continuous-  If E ⊂ ℝ, f : E → ℝ, and x₀ ∈ E, the function f is left continuous at x₀ iff for each ε > 0, there is 
δ > 0 : x₀ - δ ≤ x ≤ x₀, x ∈ E ⇒ |f(x) - f(x₀)| < ε.

Types of Discontinuities:

• Discontinuty of first kind- A function f(x) has discontinuity of first kind at x=a, if left and right hand limit exists at x=a, but are distinct. f(a+) ≠ f(a-).
• Discontinuity of second kind-A function f(x) has discontinuity of second kind at x=a, if left and right hand limits does not exists. i.e., neither f(a+) nor f(a-) exist.
• Mixed Discontinuity- A function f(x) has mixed discontinuity at x=a, if either of left or right hand limit exist.
• Removal Discontinuity- A function f(x) has the removal discontinuity at x=a, if f(a+) and f(a-) exist but f(a+) = f(a-) ≠ f(a).
• Irremoval Discontinuity- A function f(x) has irromoval discontinuity at x=a, if f has discontinuity of first kind, second kind or mixed discontinuity.
• Jumps and Jump Discontinuity- If f(a+) and f(a-) exist at x=a, then

           1. f(a) - f(a-) is called left hand jump of f at a.

           2. f(a+) - f(a) is called right hand jump of f at a.

           3. f(a+) - f(a-) is called jump of f at a.

           If any of these jumps is different from 0, then a is called the jump discontinuity of f.

           Jump discontinuity are discontinuity of first kind.

• Infinite Discontinuity- A function f(x) has infinite discontinuity at x=a, if any of the four functional limits f(a+), f(a-), suprimum of f(a+), infimum of f(a-) are indefinitely large or infinite.
• Saltus (Measure of discontinuity)- The saltus of a function f(x) at x = a is the greatest positive difference between any two of the five numbers f(a), suprimum of f(a+), infimum of f(a+), suprimum of f(a-), infimum of f(a-).
• Saltus on right- The greatest positive difference between any two of the three f(a), suprimum of f(a+), infimum of f(a+).
• Saltus on left- The greatest positive difference between any two of the three f(a), suprimum of f(a-), infimum of f(a-).
  Imp. Saltus is zero at point of continuity and greater that zero at point of discontinuity.

Sequence

Finite Sequences- A function whose domain is the first n natural number, 
i.e., {i ∈ ℕ : i≤n}.

Infinite Sequences- A function whose domain is the set of natural numbers.

Countable Set- Set is countable if it the range of some sequences.

Finite Countable Set- Set is finite countable if it is a range of come finite sequences.

Monotone Function- g :  ℕ → ℕ if  i > j  ⇒ g(i) > g(j) or g(i) < g(j).

Sequences of Real number:

Sequence- <x_n> of real number is a function whose domain is the set of natural numbers.
Limit- A real number l is a limit of a sequence <x_n>, lim x_n = l
 if ∈ > 0, ∃ N : ∀ n ≥ N, | x_n - l| < ∈.
Cauchy sequence- Sequence <x_n> is cauchy sequence if ∈ > 0, ∃ N such that
∀ n ≥ N and ∀ m ≥ N we have | x_n-x_m| < ∈.
Convergent sequence- A sequence is called convergent sequence if it has a limit.

Divergent sequence- A sequence which is not a convergent sequence is called a divergent sequence.

Cluster point- A real number l is a cluster point of the sequence  <x_n> if for given 
 ∈ > 0 and given N, ∃ n ≥ N : | x_n - l| < ∈.

Some Important Results-

1. A sequence of real number is convergent iff it is Cauchy sequence.
2. If the limit of the sequence exist it is unique.
3. Every convergent sequence is bounded.
4. If a sequence <x_n> converges to l, then its subsequence also converges to l.

Sequence-  It is a function whose domain is the set of positive integers.

                       i.e. a_n = a(n), n = 1, 2, 3, ...

Let p_n be the nth prime number,

                             p_n = p(n), n = 2, 3, 5, ...

Convergence of a sequence- A sequence {a_n} converges to real number A iff for each 

∈ > 0, there is a positive integer N, such that ∀ n ≥ N, we have | x_n - A| < ∈.

Neighbourhood- A set N_x of real numbers is a neighbourhood of a real number x iff N_x contains an interval of positive length centered at x, 

i.e., iff there is ∈ > 0 : (x -∈, x +∈) ⊂ N_x.

Accumulation point- For a set S of real numbers, a real number A is an accumulation point of S iff every neighbourhood of A contains infinitely many points of S.

Limit of a sequence- If a sequence is convergent, the unique number to which it converges is the limit of a sequence.
Subsequence- Let {a_n} be a sequence and (n_k) be any sequence of positive integers such that
n₁ < n₂ < n₃ < . . . The sequence {a_nk} is called a subsequence of {a_n} for all n = 1, 2, . . . , ∞
Increasing sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is increasing, iff a_n ≤ a_{n + 1} for all n.
Decreasing sequence- Sequence {b_n} : n = 1, 2, . . . , ∞ is decreasing, iff b_n ≥ b_{n + 1} for all n.
Monotone sequence- Sequence that is either increasing or decreasing.
Bounded above sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is bounded above, iff there exists a real number
N : a_n ≤ N for all n.
Bounded below sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is bounded below, iff there exists a real number
M : a_n ≥ M for all n.
Bounded sequence- Sequence {a_n} : n = 1, 2, . . . , ∞ is bounded, if it is bounded both from above and below
⇔ there exists a real number S : |a_n| ≤ S for all n.

Some important theorems-

1. Every convergent sequence  is a Cauchy sequence.
2. Every Cauchy sequence is bounded.
3. Every Cauchy sequence is convergent.
4. A sequence is cauchy iff it is convergent.
5. A sequence converges iff each of its subsequences converges.
6. A monotone sequence is convergent iff it is bounded.
7. If {a_n} : n = 1, 2, . . . , ∞ converges to real number A and B, then A = B.
8. If {a_n} : n = 1, 2, . . . , ∞ converges to A, then {a_n} : n = 1, 2, . . . , ∞ is bounded.

Some Important Key Points

• The measure of an angle is the amount of rotation from the initial side of the terminal side.
• The sense of an angle is positive or negative according as the initial side rotates in anticlockwise or clockwise direction to get the terminal side.
• There are three systems of measuring angles:
          (a) Sexagesimal system
          (b) Centesimal system
          (c) Circular system
• In Sexagesimal system, we have

                  1 right angle = 90 degrees (90^∘)

                  1 degree        = 60 minute (60')

                  1 minute        = 60 seconds (60")

• In centesimal system, we have

                 1 right angle = 100 grades (100^g)

                 1 grade          = 100 minutes (100')

                 1 minute        = 100 seconds (100")

• In circular system, the unit of measurement is radian. One radian is the measure of an angle subtended at the centre of a circle by an arc of length equal to the radius of the circle

                      π radian = 180^∘

• The relation between three systems

                     D/90 degree = G/100 = 2R/π

• Six trigonometric ratios

                     sin θ, cos θ, cosec θ, sec θ, tan θ, cot θ

• 1 + tan² θ = sec² θ

       or  sec θ - tan θ = 1 / (sec θ + tan θ)

• 1 + cot² θ = cosec² θ

       or cosec θ - cot θ = 1 / (cosec θ + cot θ)

•                     sin (-θ) = - sin θ,
                     cos (-θ) = cos θ,
                    tan (- θ) = tan θ,
             sin (90^∘ - θ) = cos θ,
  and    cos (90^∘ - θ) = sin θ,
             sin (90^∘+ θ) = cos θ,
  and    cos (90^∘+ θ) = - sin θ,

• Sine and cosine functions and their reciprocals i.e., cosecant and secant functions are periodic functions with period 2π whereas tangent and cotangent functions are periodic with period π.
• Odd functions: Sine, tangent, cotangent, cosecant.
• Even functions: cosine, secant.
• The curve y = tan x is a symmetric in opposite quadrants and  -∞ < y < ∞  while the curve y = sec x is symmetric about y - axis and y ≥ 1 or y ≤ -1. the value of y do not exist for
                 x = (2x + 1)π/2
• For curve y = cosec x
  and           y = cot x, y do not exist for
                   x = nπ
• Formulae
              sin (A + B) = sin A cos B + cos A sin B
               sin (A - B) = sin A cos B - cos A sin B
             cos (A + B) = cos A cos B - sin A sin B
              cos (A - B) = cos A cos B + sin A sin B
             tan (A + B) = (tan A + tan B)/ (1 - tan A tan B)
              tan (A - B) = (tan A - tan B)/ (1 + tan A tan B)
  if                  A + B = π, then
                        sin A = sin B,
                        cos A = -cos B
  and               tan A = -tan B
  and if           A + B = 2π, then
                        sin A = -sin B,
                        cos A = cos B,
  and               tan A = -tan B
• The equation a cos θ + b sin θ = c is soluable for
          |c| ≤ √(a² + b²)
• In any ΔABC, we have
                a² = b² + c² - 2bc cos A
                b² = c² + a² - 2ca cos B
                c² = a² + b² - 2ab cos C
• In any ΔABC, we have
                a = b cos C + c cos B
                b = c cos A + a cos C
                c = a cos B + b cos A
• The area Δ of a ΔABC is given by
               Δ = (1/2) bc sin A
  or         Δ = (1/2) ca sin B
  or         Δ = (1/2) ab sin C
• Let P(n) be a statement involving the natural number n such that -
  (i) P(1) is true and (ii) P(m + 1) is true, whenever P(m) is true then, P(n) is true for all n ∈ N
  This is called first principle of mathematical inductions.
• Let P(n) be a statement involving the natural number n such that -
  (i) P(1) is true and (ii) P(m + 1) is true, whenever P(n) is true for all n ≤ m,  then, P(n) is true for all n ∈ N
  This is called second principle of mathematical inductions.
• Every polynomial equation f(x) = 0 of degree n has exactly n roots real or imaginary.
• If ax² + bx + c = 0, a ≠ 0 is a quadratic equation with real coefficients, then its roots α and β given by -
                         α = (-b + √(b² -4ac))/ (2a)
                         β = (-b - √(b² -4ac))/ (2a)
  or                   α = (-b + √D)/ (2a)
                         β = (-b - √D)/ (2a)
  where           D = b² -4ac
  (i) if              D = 0, then
                        α = β = -b/2a
  so, equation has real and equal roots.
  (ii) If  D = +ve and a perfect square, then roots are rational and unequal (for a, b, c ∈ ℚ)
  for a, b, c ∈ R, D = positive and perfect square, then roots are real and distinct.
  (iii) If D > 0, but not perfect square, then roots are irrational and unequal.
  (iv) If a = 1 and b, c ∈ I, then roots are rational numbers, and roots must be integer.
• n! = 1 × 2 × 3 × 4 . . . × (n - 1) × n
• (2n)!/(n)! = 1.3.5 . . .  (2n - 1)2n
• n! + 1 is not divisible by any natural number between 2 and n.

Sets & Sets Notation

A Set is a well defined collection of objects. The term 'Well defined' means that it is possible to tell whether a given object belongs to the set or not. The students in your class from one set, the months of the year from another.

In mathematics, sets are usually denoted by capital letters as A, S or X. The objects that make up a set are referred to as its elements or members. elements of a set are normally denoted by lower-case letters such as a, s or x.

We can indicate the members of a set in various ways. If the set contains only a small number of elements, we may simply list them in any order within curly brackets. for example:

                            A= {2,4,6}

When a set contains a large or perhaps infinite number of elements and the members of the set exhibit an obvious pattern, we may indicate this pattern with three dots, which mean 'and so on' or 'and so on up to'. For example:

                            B= {3,6,9,12,...}

is the set of all positive multiples of 3, and contains an infinite number of elements, while:

                            C= {a,b,c,...,x,y,z}

is the set of letters in the alphabet.

You should familiarise yourself with the following standard notation for sets.
x ∈ A

This is read as ' x is a member of (or belongs to) A '.

Example

If V is the set of vowels in the alphabet, then the letter ' u ' is a member of V.
                                               u ∈ V

Example

If A is the set { 2, 3, 5, 7, 11 } and x = 5, then :
                                               x ∈ A
x ∉ A
This is read as ' x is not a member of (or does not belong to) A '.

Example

If V is the set of vowels in the alphabet, then the letter ' g ' is not a member of V.
                                                g ∉ V
Example

If A is the set { 2, 3, 5, 7, 11 } and x = 5 . 4, then :
                                               x ∉ A
A ⊂ B
This is read as ' A is a subset of B ' and means that every member of the set A is a member of the set B.

Example

If M is the set of months in the year and T is the set of months having 30 days, then T is a subset of M.
                                               T ⊂ M

Example

If A is the set { 3, 4, 5 } and B is the set {1, 2, 3, 4, 5, 6, 7} then:
                                               A ⊂ B
A ∪ B
This is read as ' A union B ' and is the set of all elements that are in A or in B or both.

Example

If T is the set of students who travel by train, B is the set of students who travel by bus and P is the set of students who travel by train or bus, then P is the union of sets T and B.
                                             P = T ∪ B

Example

If A is the set {a, b, c, d} and B is the set {b, d, e, f } then:
                                      A ∪ B = {a, b, c, d, e, f }

A ∩ B
This is read as ' A intersect B ' and is the set of all elements that are in both A and B.

Example

If R is the set of students who play rugby, T is the set of students who play tennis and B is the set of students who play rugby and tennis, then B is the intersection of sets R and T.
                                        B = R ∩ T

Example

If A is the set {a, b, c, d} and B is the set {b, d, e, f } then:
                                       A ∩ B = {b, d}
The main purpose of this brief introduction to set theory is to enable us to refer to various sets of numbers using the following standard notations and terminology.

N
The set of all natural numbers, also referred to as the positive integers.
                                      N = {1, 2, 3, 4, ... }

Z
This is the set of all integers
                                     Z = {..., -3, -2, -1, 0, 1, 2, 3, ... }

Q
The set of rational numbers.
A rational number is a number that can be expressed as a ratio of two integers. e.g., 1/2, -7/4, 9/2. Note that

integers are rational. e.g., 3 = 3/1.
                            
                          Q = {p/q | p ∈ Z and q ∈ N }

where the bar | is read as 'with the property' or 'such that'. There are numbers that are not rational. These numbers are called irrational numbers.
We cannot express these numbers as a ratio of two integers.
For example, √2 and π are irrational numbers.

R

The set of real numbers.
Unless you have studied complex numbers, then all numbers that you will have come across in mathematics so far are real numbers. Every point on the number line represents a real number.

[a, b]
This is the closed interval:
                          {x ∈ R | a ≤ x ≤ b}
Here a and b are real numbers and are called the endpoints. They are included in the closed interval, and this is often indicated by a filled-in dot at the ends of the interval on the number line.

(a, b)
This is the open interval:
                          {x ∈ R | a < x < b}
Again, a and b are real numbers and are called the endpionts. They are not included in the open interval and this is often indicated by an open dot at the ends of the interval on the number line.

Sometimes we can have:
                            ∞ or -∞
Note that the infinity symbols do not represent real numbers, they are used to indicate that the set is unbounded in the positive or negative direction of the real number line.
                            (-∞, b) = {x ∈ R | x < b}
                            (-∞, b] = {x ∈ R | x ≤ b}
                             (a, ∞) = {x ∈ R | x > a}
                             [a, ∞) = {x ∈ R | x ≥ a}

∃
'There exists'
e.g., 
                                   ∃ x ∈ R
such that x² = 2

∀
'For all'
e.g.,
                                  ∀ x ∈ R, x² > 0