"Theory of Functions" by Bijit Bora and Bablee Phukan is an essential textbook designed according to the FYUGP NEP syllabus for Mathematics Major students of 3rd and 5th semester under Dibrugarh University and other Indian universities. This comprehensive guide covers all key aspects of function theory with a clear and systematic approach, aiding in the conceptual understanding and practical application of mathematical principles. Published by Mahaveer Publications, this book is an excellent resource for undergraduate students preparing for advanced studies in mathematics.

I. SETS

1.1 Introduction

Set theory forms the foundation of modern mathematics and real analysis. A set is a well-defined collection of distinct objects called elements or members. In the context of real functions, we primarily work with sets of real numbers and their subsets.

1.2 Sets

1.2.1 Finite Sets

A finite set contains a limited number of elements that can be counted. Examples include:

• The set of natural numbers less than 10: {1, 2, 3, 4, 5, 6, 7, 8, 9}
• The set of roots of a polynomial equation

1.2.2 Countable Sets

A set is countable if its elements can be put in one-to-one correspondence with the natural numbers. This includes:

• The set of integers ℤ
• The set of rational numbers ℚ
• Any subset of a countable set

1.2.3 Uncountable Sets

Sets that cannot be put in correspondence with natural numbers are uncountable. The most important example is:

• The set of real numbers ℝ
• The set of irrational numbers

1.2.4 Bounded and Unbounded Sets

• Bounded sets: A set S ⊆ ℝ is bounded if there exists M > 0 such that |x| ≤ M for all x ∈ S
• Unbounded sets: Sets that are not bounded, extending infinitely in at least one direction

1.3 The Set of Real Numbers ℝ

The real numbers form a complete ordered field with the least upper bound property. They include:

• Natural numbers ℕ = {1, 2, 3, ...}
• Integers ℤ = {..., -2, -1, 0, 1, 2, ...}
• Rational numbers ℚ
• Irrational numbers

1.4 Algebraic Properties of ℝ

The real numbers satisfy:

• Addition: Associative, commutative, with identity 0 and inverses
• Multiplication: Associative, commutative, with identity 1 and inverses (except 0)
• Distributive law: a(b + c) = ab + ac

1.5 Order Properties of ℝ

1.5.1 The Order Structure ≤ of ℝ

For any a, b ∈ ℝ, exactly one of the following holds:

• a < b, a = b, or a > b (trichotomy)
• If a ≤ b and b ≤ c, then a ≤ c (transitivity)

1.5.2-1.5.4 Consequences and Completeness

The completeness property ensures every non-empty bounded set has a supremum and infimum.

1.6 Density Theorem

Between any two distinct real numbers, there exists both a rational and an irrational number.

1.7 Intervals

Types of intervals:

• Open: (a, b) = {x ∈ ℝ : a < x < b}
• Closed: [a, b] = {x ∈ ℝ : a ≤ x ≤ b}
• Half-open: [a, b), (a, b]

1.8 Absolute Value

|x| = x if x ≥ 0, |x| = -x if x < 0

Properties:

• |xy| = |x||y|
• |x + y| ≤ |x| + |y| (Triangle inequality)

1.9 Functions

A function f: A → B assigns to each element x ∈ A exactly one element f(x) ∈ B.

1.10 Composition of Functions

If f: A → B and g: B → C, then (g ∘ f)(x) = g(f(x))

1.10.1 Inverse Functions

f⁻¹ exists if f is bijective (one-to-one and onto)

1.11 Sequences

1.11.1 Limit of a Sequence

A sequence {aₙ} converges to L if for every ε > 0, there exists N such that |aₙ - L| < ε for all n > N.

1.11.2 Bounded Sequences

A sequence is bounded if there exists M > 0 such that |aₙ| ≤ M for all n.

1.11.3 Convergence

Every convergent sequence is bounded. The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence.

1.12 Algebra of Sequences

If {aₙ} → a and {bₙ} → b, then:

• {aₙ + bₙ} → a + b
• {aₙbₙ} → ab
• {aₙ/bₙ} → a/b (if b ≠ 0)

1.13 Subsequences

1.13.1 Cauchy Sequences

A sequence {aₙ} is Cauchy if for every ε > 0, there exists N such that |aₘ - aₙ| < ε for all m, n > N. In ℝ, Cauchy sequences converge (completeness).

II. LIMITS OF FUNCTIONS

2.1 Introduction

Limits describe the behavior of functions near specific points and form the foundation for continuity and derivatives.

2.2 Real Functions

A real function is a mapping f: D → ℝ where D ⊆ ℝ is the domain.

2.3 Neighborhood

An ε-neighborhood of point a is the open interval (a - ε, a + ε).

2.4 Deleted Neighborhood

The set (a - ε, a + ε) \ {a}, excluding the point a itself.

2.5 Cluster Point (Limit Point)

A point a is a cluster point of set S if every neighborhood of a contains points of S other than a itself.

2.6 Limit of a Function

lim[x→a] f(x) = L if for every ε > 0, there exists δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε.

2.7 Sequential Criterion for Limits

lim[x→a] f(x) = L if and only if for every sequence {xₙ} → a (xₙ ≠ a), we have f(xₙ) → L.

2.8 Divergence Criteria

A function diverges if the limit doesn't exist or is infinite.

2.9 One-Sided Limits

• Right limit: lim[x→a⁺] f(x)
• Left limit: lim[x→a⁻] f(x)

2.10 Infinite Limits

lim[x→a] f(x) = ∞ if for every M > 0, there exists δ > 0 such that 0 < |x - a| < δ implies f(x) > M.

2.11 Limits at Infinity

lim[x→∞] f(x) = L if for every ε > 0, there exists M such that x > M implies |f(x) - L| < ε.

III. CONTINUOUS FUNCTIONS

3.1 Introduction

Continuity formalizes the intuitive notion of functions without "jumps" or "breaks."

3.2 Continuous Functions

f is continuous at a if lim[x→a] f(x) = f(a).

3.2.1 Sequential Criterion for Continuity

f is continuous at a if and only if for every sequence {xₙ} → a, we have f(xₙ) → f(a).

3.2.2 Discontinuity Criteria

Types of discontinuities:

• Removable: limit exists but ≠ f(a)
• Jump: left and right limits exist but are different
• Essential: at least one one-sided limit doesn't exist

3.3 Types of Discontinuity

Classification based on behavior of one-sided limits.

3.4 Continuous Functions on Intervals

Properties of functions continuous on closed intervals.

3.5 Maximum-Minimum Theorem

Every continuous function on a closed bounded interval attains its maximum and minimum values.

3.6 Location of Roots Theorem

If f is continuous on [a, b] and f(a)f(b) < 0, then there exists c ∈ (a, b) such that f(c) = 0.

3.7 Bolzano's Intermediate Value Theorem

If f is continuous on [a, b] and k is between f(a) and f(b), then there exists c ∈ (a, b) such that f(c) = k.

3.8 Preservation of Intervals Theorem

The continuous image of an interval is an interval.

3.9 Uniform Continuity

f is uniformly continuous on D if for every ε > 0, there exists δ > 0 such that |x - y| < δ implies |f(x) - f(y)| < ε for all x, y ∈ D.

3.10 Non-uniform Continuity Criteria

Methods to show a function is not uniformly continuous.

3.11 Uniform Continuity Theorem

Every continuous function on a closed bounded interval is uniformly continuous.

3.12 Lipschitz Functions

f satisfies a Lipschitz condition if |f(x) - f(y)| ≤ K|x - y| for some constant K.

3.12.1 Geometrical Interpretation

Lipschitz functions have bounded slope everywhere.

IV. DIFFERENTIATION

4.1 Introduction

Differentiation measures instantaneous rates of change and slopes of tangent lines.

4.2 Derivative

f'(a) = lim[h→0] [f(a+h) - f(a)]/h, provided the limit exists.

4.3 Algebra of Differential Functions

Rules for derivatives:

• (f + g)' = f' + g'
• (fg)' = f'g + fg' (Product rule)
• (f/g)' = (f'g - fg')/g² (Quotient rule)
• (f ∘ g)' = f'(g(x))g'(x) (Chain rule)

4.4 Carathéodory's Theorem

f is differentiable at a if and only if there exists a function φ continuous at a such that f(x) - f(a) = φ(x)(x - a).

V. APPLICATION OF DERIVATIVES

5.1 Introduction

Derivatives have numerous applications in optimization, curve sketching, and physics.

5.2 Rolle's Theorem

If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) such that f'(c) = 0.

5.3-5.4 Mean Value Theorems

Geometrical Interpretation

The mean value theorem guarantees a point where the tangent is parallel to the secant line.

Lagrange's Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that f'(c) = [f(b) - f(a)]/(b - a).

5.5 Geometrical Interpretation of Mean Value Theorem

Visual understanding of why the theorem holds.

5.6 Increasing Functions

f is increasing on I if f'(x) ≥ 0 for all x ∈ I.

5.7 Decreasing Functions

f is decreasing on I if f'(x) ≤ 0 for all x ∈ I.

5.8 Darboux's Theorem (Intermediate Value Theorem for Derivatives)

If f is differentiable on [a, b] and k is between f'(a) and f'(b), then there exists c ∈ (a, b) such that f'(c) = k.

5.9 Cauchy Mean Value Theorem

Generalization of the mean value theorem for ratios of functions.

5.10 Geometrical Interpretation of Cauchy Mean Value Theorem

Understanding through parametric curves.

5.11 Taylor's Theorem

Approximation of functions using polynomials: f(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + Rₙ(x)

5.12 Taylor's Theorem with Cauchy Form of Remainder

Specific form of the remainder term in Taylor's theorem.

5.13 Lagrange's Form of Remainder

Another expression for the remainder in Taylor expansions.

5.14 Cauchy's Form of Remainder

Alternative remainder formula.

5.15 Taylor's Series About a Point

Infinite series representation of functions.

5.16 Maclaurin's Series (Taylor's Series About the Origin)

Special case where expansion is around x = 0.

5.17 Convex Functions

f is convex if f(λx + (1-λ)y) ≤ λf(x) + (1-λ)f(y) for 0 ≤ λ ≤ 1.

5.17.1 Concave Functions

f is concave if -f is convex.

5.18 Relative Extrema or Local Extrema

Points where functions achieve local maximum or minimum values.

5.18.1 First Derivative Test for Extrema

If f'(c) = 0 and f' changes sign at c, then f has a local extremum at c.

5.18.2 Second Derivative Test

If f'(c) = 0 and f''(c) > 0, then f has a local minimum at c.

Key Theorems and Applications

Fundamental Theorems:

1. Bolzano-Weierstrass Theorem: Every bounded sequence has a convergent subsequence
2. Intermediate Value Theorem: Continuous functions take on all intermediate values
3. Extreme Value Theorem: Continuous functions on closed intervals attain extrema
4. Mean Value Theorem: Foundation for many applications in calculus
5. Taylor's Theorem: Basis for function approximation and analysis