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Function see math notation, mean value theorem, intermediate value theorem definition a function `HH` between sets `A` and `B` is a relation between `A` and `B` such that: `HH a b /\ B b > A a` or alternatively `B h a > A a` --- there exists some output for every input `HH a b_1 /\ HH a b_2 < b_1 = b_2` or alternatively `a_1 = a_2 < h a_1 = h a_2` --- there exists exactly one output for any input one can use the horizontal line test to determine whether the graph of a curve is the graph of a function types boolean function trigonometric function hyperbolic function predicate applications functional programming function› vector space activation function loss function Function Vector Space see vector, vector space definition `HH f` if and only if the function `f` is defined on its whole function › domain #xxx if and only if it is a set theoryetical function? properties zero function `O x = 0` function addition `(f : g) x = f x : g x` multiplication by a scalar `cf x = c \| f x` Domain Codomain Range Root definition the domain of a function is the set of arguments for which it will produce an output definition the codomain of a function is the set of all outputs it may or may not produce definition the range of a function is the set of all outputs it definitely can produce definition the roots of a function `f` are the values of `x` such that `f x = 0` this asymmetry between the "input" and the "output" of a function is what distinguishes it from a relation --- https://youtu.be/O2lZkr-aAqk?t=724 properties codomain `D x < C f x` range `D x == R f x` `D x > A f x` Parity Even Function an even function is symmetrical about the y axis definition `f x = f ..x > RR x` Odd Function an odd function is symmetrical about the y axis, but also flipped about the x axis definition `..f x = f ..x > RR x` Periodic Function definition `f x = f (x : p) /\ RR p > RR x` definition above, `p` is said to be the period of `f` Multivalued Function definition a function is multivalued if it maps a single input to multiple outputs #todo link with improved expression evaluation superpositions #xxx aren't those "superpositions" just the List monad (non-determinism)? Increasing Function Decreasing Function see calculus notation definition a function `f` is increasing on an interval *`a -\| -\| b` if `x_1 -\| x_2 < f x_1 -\| f x_2`, or `dd f \|- 0` on that interval definition a function `f`** is decreasing on an interval *`a -\| -\| b` if `x_1 \|- x_2 < f x_1 \|- f x_2`, or `dd f -\| 0` on that interval Concavity see calculus notation definition a function `f`** is concave up at `x` if `dd2 f x \|- 0`; it bends upwards definition a function `f` is concave down at `x` if `dd2 f x -\| 0`; it bends downwards a point where function › concavity changes (from up to down or down to up) is a function › inflection point Extremum see function › inflection point, derivative extrema are the largest and smallest value of the function, either within a given range (the local or relative extrema), or on the entire function › domain (the global or absolute extrema). --- Wikipedia definition the global extrema `x` of a function `f` with function › domain `D` are defined as `f x \|- f y > D y` and `f x -\| f y > D y` definition the global extrema of a function are the absolute highest and lowest points of the function definition the local extrema of a function are the highest and lowest points of the function within a given range theorem if `f` has a local function › extremum at `c`, the point `(c, f c)` is a function › critical point of `f`, but not conversely First Derivative Test let `f` be a continuous ‹function near `x = c` and `c` be a critical number of `f`. then, `f` has a local function › extremum at `c` if `dd f` changes sign at `c` Second Derivative Test let `f` be a continuous ‹function near `x = c` and `c` be a critical number of `f` where `dd f c = 0`. then, `f` has a local maximum at `c` if `dd2 f c -\| 0` a local minimum at `c` if `dd2 f c \|- 0` note the test is inconclusive if `dd2 f c = 0` or if it does not exist Inflection Point see function › extremum definition An inflection point [...] is a point on a smooth plane curve at which the function › curvature changes sign --- Wikipedia definition a function `f` has an inflection point at `c` if it is continuous at `c` and its function › concavity changes sign at `c` note a function having its second derivative equal to zero at `c` does not imply `c` is a function › inflection point. as an example, `f x = x4` does not have a function › inflection point at `(x, f x) = (0, 0)` Critical Point see function, math notation definition function of one variable a point `(c, f c)` is a critical point of the function `f` if `dd f c = 0` or it does not exist definition function of multiple variables a point `(c, f c)` is a critical point of the function `f` if all components of `dd f c` either are zero or do not exist definition above, `c` would be said to be a critical number of `f` Continuous Function see math notation definition a function `f x` is continuous at `x = a` if `f x {x -> a} = f a` #todo lim note above, `f x {x -> a}` #todo lim must exist and `f x` must be defined at `x = a` definition a function is continuous from the left at `a` when `f x {x -> a "from the left"} = f a` #todo lim and both other conditions are met definition a function is continuous from the right at `a` when `f x {x -> a "from the right"} = f a` #todo lim and both other conditions are met definition a function is continuous on an interval *`a -\| -\| b` if it is continuous on every point from `a` to `b` exclusively, and continuous from the right at `a` and from the left at `b` definition** a function is continuous (globally continuous) if it is continuous on every point of its function › domain theorem if `f x` and `g x` are continuous at `a`, then the following functions are also continuous at `a`: `f : g` `f .. g` `f \| g` `c ' f` where `c` is a scalar `f -- g` if `g a + 0` (restriction not necessary, see improved expression evaluation) Slope definition `dd f` Tangent Line used to represent function › slope at a point a tangent line has the same function › slope as a given function at a point definition `L x = f a : (x..a \| dd f a)`, where `L` is the line tangent to `f` at `a` applications the tangent of a function `f` approximates `f (x ...)` near a point `(x ...)` function › tangent lines are used in newton's method Curvature definition `dd2 f` Osculating Circle used to represent function › curvature at a point --- https://youtu.be/jvPPXbo87ds?t=1847 an osculating circle has the same function › slope and the same function › curvature as a given function at a point Average see integral definition `"ave" f = $ f {b..a} -- b .. a`, where `$ f` is an antiderivative of `f` with respect to `x` `"ave" f` is the average of the function `f` on the interval *`a -\| -\| b` --- https://youtu.be/7gigNsz4Oe8?t=3093 --- https://youtu.be/FnJqaIESC2s Arclength see integral definition `"arc" f = $ t. "abs" (dd f t)` where `f t = (x, f x ...)` --- https://tutorial.math.lamar.edu/classes/calciii/vectorarclength.aspx definition `"arc" f = $ x. "abs" (1, dd f x)`, see euclidean vector › magnitude Injective Function aka** one-to-one function definition a function `f` is said to be injective if `f x_1 = f x_2 < x_1 = x_2`, see universal ‹set. for every output value there exists at most one input mapping to it given the graph of a function, one can use the horizontal line test to determine whether it is injective or not a function can be proven to be injective by proving that two output values being equal implies that the corresponding input values are equal Surjective Function aka onto function definition a function `f` with function › codomain `C` is said to be surjective if `C y < f x = y`, see universal ‹set. for every output value there exists at least one input mapping to it a function can be proven to be surjective by proving one can construct an input value for the function given an arbitrary output value example proving a function is surjective let `y = f m n = m : n`. then, suppose `m = 0`. solving for `n`, we get `n = y`. therefore, the function is surjective example proving a function is not surjective let `y = f m n = m2 : n2`. `y = ..1` would cause a contradiction as the square of an integer is always a positive integer and the sum of two positive integers is always a positive integer. therefore, the function is not surjective example proving a function is surjective let `y = f m n = m`. then, we get `m = y` and therefore the function is surjective example proving a function is not surjective let `y = f m n = "abs" n`. `y = ..1` would cause a contradiction as the real› absolute value of an integer is always a positive integer. therefore, the function is not surjective example proving a function is surjective let `y = f m n = m .. n`. then, suppose `n = 0`. solving for `m`, we get `m = y`. therefore, the function is surjective Bijective Function definition a function `f` is said to be bijective if it is both injective and surjective. for every output value there exists exactly one input mapping to it a function can be proven to be bijective by proving it is both injective and surjective Analytic Function see derivative definition an analytic function is a function that is locally given by a convergent ‹series power ‹series --- Wikipedia properties a analytic ‹function is infinitely differentiable, but an infinitely differentiable function is not necessarily analytic --- https://youtu.be/X0razs3zR94?t=598 analytic continuation of a analytic ‹function is uniquely determined --- https://youtu.be/YuIIjLr6vUA?t=1746 Partial Function Total Function --- https://wiki.haskell.org/Partial_functions --- https://www.parsonsmatt.org/2017/10/11/type_safety_back_and_forth.html by Matt Parson definition a partial function maps a sub‹set of its function › domin to some element of its function › codomain definition a total function maps every element of its function › domain to some element of its function › codomain in computer science and functional programming, total functions are functions that don't "lie" in their type signature. a total function never `panic!`s, never throw exceptions, and always terminates example `fn reciprocal(x: f64) -> f64 { 1 / x }` the function has type `fn(f64) -> f64`, yet `f(0.0)` is undefined. you could add an `if` check and throw an exception, but the type signature would still be a lie. in functional programming, one should prefer either: restricting the input of the function, `fn(NonZeroF64) -> f64` extending the output of the function, `fn(f64) -> Option<f64>`
© 2026 Emilien Breton ← 🍁 →

Function

see math notation, mean value theorem, intermediate value theorem

definition

a function HH between sets A and B is a relation between A and B such that:

HH a b /\ B b > A a or alternatively B h a > A a --- there exists some output for every input
HH a b_1 /\ HH a b_2 < b_1 = b_2 or alternatively a_1 = a_2 < h a_1 = h a_2 --- there exists exactly one output for any input

one can use the horizontal line test to determine whether the graph of a curve is the graph of a function

types

boolean function

trigonometric function

hyperbolic function

predicate

applications

functional programming

function› vector space

activation function

loss function

Function Vector Space

see vector, vector space

definition HH f if and only if the function f is defined on its whole function › domain

#xxx if and only if it is a set theoryetical function?

properties

zero function O x = 0

function addition (f : g) x = f x : g x

multiplication by a scalar cf x = c | f x

Domain

Codomain

Range

Root

definition the domain of a function is the set of arguments for which it will produce an output

definition the codomain of a function is the set of all outputs it may or may not produce

definition the range of a function is the set of all outputs it definitely can produce

definition the roots of a function f are the values of x such that f x = 0

this asymmetry between the "input" and the "output" of a function is what distinguishes it from a relation --- https://youtu.be/O2lZkr-aAqk?t=724

properties

codomain D x < C f x

range D x == R f x

D x > A f x

Parity

Even Function

an even function is symmetrical about the y axis

definition f x = f ..x > RR x

Odd Function

an odd function is symmetrical about the y axis, but also flipped about the x axis

definition ..f x = f ..x > RR x

Periodic Function

definition f x = f (x : p) /\ RR p > RR x

definition above, p is said to be the period of f

Multivalued Function

definition a function is multivalued if it maps a single input to multiple outputs

#todo link with improved expression evaluation superpositions

#xxx aren't those "superpositions" just the List monad (non-determinism)?

Increasing Function

Decreasing Function

see calculus notation

Concavity

see calculus notation

definition a function f is concave up at x if dd2 f x |- 0; it bends upwards

definition a function f is concave down at x if dd2 f x -| 0; it bends downwards

a point where function › concavity changes (from up to down or down to up) is a function › inflection point

Extremum

see function › inflection point, derivative

extrema are the largest and smallest value of the function, either within a given range (the local or relative extrema), or on the entire function › domain (the global or absolute extrema). --- Wikipedia

definition

the global extrema x of a function f with function › domain D are defined as

f x |- f y > D y and f x -| f y > D y

definition the global extrema of a function are the absolute highest and lowest points of the function

definition the local extrema of a function are the highest and lowest points of the function within a given range

theorem if f has a local function › extremum at c, the point (c, f c) is a function › critical point of f, but not conversely

First Derivative Test

let f be a continuous ‹function near x = c and c be a critical number of f. then, f has a local function › extremum at c if dd f changes sign at c

Second Derivative Test

let f be a continuous ‹function near x = c and c be a critical number of f where dd f c = 0. then, f has

a local maximum at c if dd2 f c -| 0
a local minimum at c if dd2 f c |- 0

note the test is inconclusive if dd2 f c = 0 or if it does not exist

Inflection Point

see function › extremum

definition An inflection point [...] is a point on a smooth plane curve at which the function › curvature changes sign --- Wikipedia

definition a function f has an inflection point at c if it is continuous at c and its function › concavity changes sign at c

note a function having its second derivative equal to zero at c does not imply c is a function › inflection point. as an example, f x = x4 does not have a function › inflection point at (x, f x) = (0, 0)

Critical Point

see function, math notation

definition function of one variable a point (c, f c) is a critical point of the function f if dd f c = 0 or it does not exist

definition function of multiple variables a point (c, f c) is a critical point of the function f if all components of dd f c either are zero or do not exist

definition above, c would be said to be a critical number of f

Continuous Function

see math notation

definition a function f x is continuous at x = a if f x {x -> a} = f a #todo lim

note above, f x {x -> a} #todo lim must exist and f x must be defined at x = a

definition a function is continuous from the left at a when f x {x -> a "from the left"} = f a #todo lim and both other conditions are met

definition a function is continuous from the right at a when f x {x -> a "from the right"} = f a #todo lim and both other conditions are met

definition a function is continuous on an interval a -| * -| b if it is continuous on every point from a to b exclusively, and continuous from the right at a and from the left at b

definition a function is continuous (globally continuous) if it is continuous on every point of its function › domain

theorem

if f x and g x are continuous at a, then the following functions are also continuous at a:

f : g
f .. g
f | g
c ' f where c is a scalar
f -- g if g a + 0 (restriction not necessary, see improved expression evaluation)

Slope

definition dd f

Tangent Line

used to represent function › slope at a point

a tangent line has the same function › slope as a given function at a point

definition

L x = f a : (x..a | dd f a), where

L is the line tangent to f at a

applications

the tangent of a function f approximates f (x ...) near a point (x ...)

function › tangent lines are used in newton's method

Curvature

definition dd2 f

Osculating Circle

used to represent function › curvature at a point

--- https://youtu.be/jvPPXbo87ds?t=1847

an osculating circle has the same function › slope and the same function › curvature as a given function at a point

Average

see integral

definition

"ave" f = $ f {b..a} -- b .. a, where

$ f is an antiderivative of f with respect to x
"ave" f is the average of the function f on the interval a -| * -| b

--- https://youtu.be/7gigNsz4Oe8?t=3093

--- https://youtu.be/FnJqaIESC2s

Arclength

see integral

definition "arc" f = $ t. "abs" (dd f t) where f t = (x, f x ...) --- https://tutorial.math.lamar.edu/classes/calciii/vectorarclength.aspx

definition "arc" f = $ x. "abs" (1, dd f x), see euclidean vector › magnitude

Injective Function

aka one-to-one function

definition a function f is said to be injective if f x_1 = f x_2 < x_1 = x_2, see universal ‹set. for every output value there exists at most one input mapping to it

given the graph of a function, one can use the horizontal line test to determine whether it is injective or not

a function can be proven to be injective by proving that two output values being equal implies that the corresponding input values are equal

Surjective Function

aka onto function

definition a function f with function › codomain C is said to be surjective if C y < f x = y, see universal ‹set. for every output value there exists at least one input mapping to it

a function can be proven to be surjective by proving one can construct an input value for the function given an arbitrary output value

example proving a function is surjective

let y = f m n = m : n. then, suppose m = 0. solving for n, we get n = y. therefore, the function is surjective

example proving a function is not surjective

let y = f m n = m2 : n2. y = ..1 would cause a contradiction as the square of an integer is always a positive integer and the sum of two positive integers is always a positive integer. therefore, the function is not surjective

example proving a function is surjective

let y = f m n = m. then, we get m = y and therefore the function is surjective

example proving a function is not surjective

let y = f m n = "abs" n. y = ..1 would cause a contradiction as the real› absolute value of an integer is always a positive integer. therefore, the function is not surjective

example proving a function is surjective

let y = f m n = m .. n. then, suppose n = 0. solving for m, we get m = y. therefore, the function is surjective

Bijective Function

definition a function f is said to be bijective if it is both injective and surjective. for every output value there exists exactly one input mapping to it

a function can be proven to be bijective by proving it is both injective and surjective

Analytic Function

see derivative

definition an analytic function is a function that is locally given by a convergent ‹series power ‹series --- Wikipedia

properties

a analytic ‹function is infinitely differentiable, but an infinitely differentiable function is not necessarily analytic --- https://youtu.be/X0razs3zR94?t=598

analytic continuation of a analytic ‹function is uniquely determined --- https://youtu.be/YuIIjLr6vUA?t=1746

Partial Function

Total Function

--- https://wiki.haskell.org/Partial_functions

--- https://www.parsonsmatt.org/2017/10/11/type_safety_back_and_forth.html by Matt Parson

definition a partial function maps a sub‹set of its function › domin to some element of its function › codomain

definition a total function maps every element of its function › domain to some element of its function › codomain

in computer science and functional programming, total functions are functions that don't "lie" in their type signature. a total function never panic!s, never throw exceptions, and always terminates

example
fn reciprocal(x: f64) -> f64 {
  1 / x
}
the function has type fn(f64) -> f64, yet f(0.0) is undefined. you could add an if check and throw an exception, but the type signature would still be a lie. in functional programming, one should prefer either:

restricting the input of the function, fn(NonZeroF64) -> f64

extending the output of the function, fn(f64) -> Option<f64>