﻿ ClosedInterval – Algosim documentation
Algosim documentation: ClosedInterval

# ClosedInterval

Returns a closed interval of real numbers.

## Syntax

• `ClosedInterval(a, b[, δ])`

• `a` and `b` are real numbers

• `δ` is a positive number

## Description

Conceptually, `ClosedInterval(a, b)` returns the closed interval [a, b]. However, there are a few differences between the mathematical object [a, b] and `ClosedInterval(a, b)`.

First, since the mathematical set [a, b] contains an infinite number of real numbers when a < b (the typical case), it cannot be represented as a true Algosim set, since that would require a computer with an infinite amount of memory. Therefore, `ClosedInterval(a, b)` only returns a finite number of equally spaced numbers from this interval.

Second, the returned object isn’t an Algosim set but a list. This is because lists are easier to work with in practice. Still, Algosim is designed in such a way that you often can think of `ClosedInterval(a, b)` as the mathematical object [a, b].

Specifically, `CloseInterval(a, b, δ)` returns the list of all numbers

`a, a + δ, a + 2⋅δ, ...`

that are less than or equal to `b`. In particular, if `a > b`, the empty list is returned and if `a = b`, the single-element list containing only `a` is returned. Otherwise, the number of elements depends on `δ`. If omitted, `δ` is chosen so that the list contains about 1000 elements.

A synonym for `ClosedInterval` is `interval`. Also, the circumfix operator `[]` is mapped to this function. Hence, `ClosedInterval(a, b)` can also be written `interval(a, b)` or `[a, b]`.

## Examples

```f ≔ t ↦ t⋅❨cos(t), sin(t)❩;
plot([0, 6⋅π] @ f)
``` ```f ≔ t ↦ (e^sin(t) − 2⋅cos(4⋅t) + sin((2⋅t − π)/24)^5) ⋅ ❨cos(t), sin(t)❩;
plot([0, 100, 0.01] @ f)
``` 