Numbers and the Real Number System

Types of Numbers

The real number system is foundational to calculus, encompassing several subsets with distinct properties. Understanding these sets is essential for grasping limits, continuity, and other calculus concepts.

• Natural Numbers (ℕ): The set of counting numbers, typically {0, 1, 2, 3, ...}. Defined by the Peano Axioms, which establish properties such as the existence of a unique successor for each number and the principle of induction.

• Integers (ℤ): Includes all whole numbers, both positive and negative, and zero: {..., -3, -2, -1, 0, 1, 2, 3, ...}. Each integer has both a successor and a predecessor.

• Rational Numbers (ℚ): Numbers expressible as the ratio of two integers, , where . Rational numbers have both finite and repeating decimal representations.

• Real Numbers (ℝ): Includes all rational and irrational numbers. Real numbers can be constructed as limits of sequences of rationals (Cauchy sequences).

Hierarchy:

Key Properties

• Ordering: For any , exactly one of , , or holds.

• Density: Between any two distinct rationals (or reals), there exists another rational (or real).

• Completeness: Every non-empty subset of that is bounded above has a least upper bound (supremum).

Intervals and Bounds

Subsets of are often described as intervals, which may be bounded or unbounded.

• Open Interval:

• Closed Interval:

• Half-Open Intervals: or

• Unbounded Intervals: , , ,

Definitions:

• Maximum: is the maximum if for all .

• Upper Bound: is an upper bound if for all .

• Supremum: is the least upper bound if is an upper bound and for any upper bound .

• Bounded: is bounded if such that for all .

Table: Types of Intervals

Limits of Sequences

Sequences and Their Limits

A sequence is an ordered list of numbers, often denoted . The concept of a limit is central to calculus, describing the behavior of sequences as becomes large.

• Definition: A sequence converges to if for every , such that for all .

• Divergence: If no such exists, the sequence diverges. Divergence may be to , , or oscillatory.

Examples:

• converges to $0$.

• diverges to .

• diverges (oscillates between $1-1$).

• Fibonacci sequence: , , for .

Limits of Functions

Definition of Limit at a Point

The limit of a function as approaches describes the behavior of $f(x)$ near $x_0$.

• Formal Definition: if for every , such that whenever .

• One-sided Limits: (from the right), (from the left).

• Infinite Limits: if grows without bound as .

• Limits at Infinity: if approaches as increases without bound.

Examples:

• does not exist

Continuity

Intuitive and Formal Definitions

Continuity describes functions whose graphs can be drawn without lifting the pen, but the formal definition is more precise.

• Continuity at a Point: is continuous at if for every , such that whenever and .

• Continuous Function: is continuous if it is continuous at every point in its domain.

Theorems:

• Continuity Implies Limit: If is continuous at , then .

• Limit Implies Continuity: If , then is continuous at .

Types of Discontinuities

• Removable Discontinuity: A 'hole' in the graph; can be fixed by redefining .

• Jump Discontinuity: The left and right limits exist but are not equal.

• Oscillating Discontinuity: The function oscillates near and the limit does not exist.

• Infinite Discontinuity: The function approaches or near .

Examples:

• has a removable discontinuity at .

• has an infinite discontinuity at .

• has a jump discontinuity at .

Topology and Continuity

Open and Closed Sets

Topology provides a framework for understanding continuity in terms of open sets.

• Open Ball:

• Interior Point: is an interior point of if such that .

• Open Set: is open if every point is an interior point.

• Limit Point: is a limit point of if every open ball around $x_0$ contains a point of $\Omega$ other than $x_0$.

• Closed Set: is closed if it contains all its limit points.

• Isolated Point: is isolated if some open ball around $x_0$ contains no other points of .

Topological Definition of Continuity

A function is continuous if the pre-image of every open set is open.

• Image:

• Pre-Image:

Properties of Continuous Functions

Algebraic Operations

Continuous functions retain continuity under various operations:

• Sum:

• Difference:

• Product:

• Quotient: (where )

• Scalar Multiplication: for

• Powers: ,

• Composition:

Examples:

• Polynomials are continuous everywhere.

• Rational functions are continuous except at points where the denominator is zero.

Summary Table: Types of Discontinuities

Additional info:

• Some handwritten notes and images referenced the Squeeze Theorem and tangent lines, which are foundational for later calculus topics such as derivatives.

• Problems and exercises included in the material reinforce understanding of limits, continuity, and properties of real numbers.