1.B. Intervals - JulTob/Mathematics GitHub Wiki

Intervals in Real Analysis πŸ“πŸ§ βœ¨

Intervals are fundamental constructs in mathematics used to describe subsets of the real line $( \mathbb{R} )$. They define continuous ranges of real numbers bounded by two endpoints, typically labeled $( a )$ and $( b )$, where $( a < b )$. Each type of interval includes or excludes these boundaries in different ways.

Closed Interval [a, b] πŸ”’πŸ“

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β”„β”„β”„β•Šβ”β”β”β”β”β”β”β•‰β”„β”„
  a         b

Definition:

 [a, b] = \{ x \in \mathbb{R} \mid a \leq x \leq b \}
  • Includes both endpoints $( a )$ and $( b )$
  • Represents all real numbers from $( a )$ to $( b )$, inclusive
  • Compact and bounded β€” important in topology and calculus

Example: All real numbers between 2 and 5, including 2 and 5.

Open Interval (a, b) πŸ”“πŸŒ€

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┄┄┄┾━━━━━━━┽┄┄
  a         b

Definition:

 (a, b) = \{ x \in \mathbb{R} \mid a < x < b \}
  • Excludes endpoints
  • Only the interior points are included
  • Often used to define neighborhoods around a point

Alternate Notation: $( ]a, b[ )$ (common in European texts)

Semi-Open (Half-Open) Intervals βš–οΈβ†”οΈ

These intervals include one endpoint but exclude the other.

Left-Closed, Right-Open: [a, b)

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β”„β”„β”„β•Šβ”β”β”β”β”β”½β”„β”„
  a       b

Definition:

 [a, b) = \{ x \in \mathbb{R} \mid a \leq x < b \}
  • Includes $( a )$, excludes $( b )$
  • Used in function domains and Riemann integration

Alternate Notation: $( [a, b[ )$

Left-Open, Right-Closed: (a, b]

⬜️⬜️🟒🟩🟩🟩🟩βšͺ️◻️◽️▫️
┄┄┄┾━━━━━━━╉┄┄
  a         b

Definition:

 (a, b] = \{ x \in \mathbb{R} \mid a < x \leq b \}
  • Excludes $( a )$, includes $( b )$
  • Also appears frequently in step functions or limit constructions

Alternate Notation: $( ]a, b] )$

Intervals and the Real Line πŸ”’πŸ“Š

Think of $( a )$ and $( b )$ as specific points on a real number line:

 P = \{ x_0, x_1, x_2, \dots, x_n \} \text{ where } a = x_0, b = x_n

This set defines the possible values within the range determined by $( a )$ and $( b )$, either including or excluding boundaries depending on the interval type.

Why Intervals Matter πŸ§ πŸ’‘

  • Continuity: Intervals are domains for continuous functions.
  • Limits: Open intervals are crucial in the $( \varepsilon-\delta )$ definition of limits.
  • Integration: Riemann sums and integrals are taken over closed or semi-open intervals.
  • Topology: Open and closed intervals are basic building blocks of topological spaces.

πŸ“ Infinite Intervals

Let a be any real number. The set of all numbers x such that:

x < a

is called an infinite interval. Other forms of infinite intervals include:

  • $x \leq a$
  • $x > a$
  • $x \geq a$

These intervals extend indefinitely in one direction and are fundamental in understanding unbounded domains of variation.

πŸ” Constants and Variables

In an expression like:

a < x < b
  • The symbols $a$ and $b$ each represent a fixed number. These are called constants.
  • The symbol $x$ represents any number that lies between $a$ and $b$. This is known as a variable.

The range of variation of a variable refers to the set of all possible values it can take. This set defines the β€œuniverse” or β€œdomain” the variable lives in.

Examples:

  1. Books in a series

    • If $x$ represents a book in a series of 10 volumes, then the range of variation for x is: ${1, 2, 3, \ldots, 10}$

  2. Days in July

    • If $x$ represents a day in the month of July, its range of variation is: ${1, 2, 3, \ldots, 31}$

  3. Water in a tank

    • If $x$ is the amount of water (in liters) drawn from a full tank of 10 liters, then the variation range is the interval: $0 \leq x \leq 10$

Final Note πŸ«πŸ›ŒπŸ’€

Now you know the types of intervals β€” what they look like, what they mean, and why they matter. That’s enough brain work for now. Go rest, have something sweet, and return recharged. 😊