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In mathematics, an injective function (also known as injection, or one-to-one function [1]) is a function f that maps distinct elements of its domain to distinct elements of its codomain; that is, x 1 ≠ x 2 implies f(x 1) ≠ f(x 2) (equivalently by contraposition, f(x 1) = f(x 2) implies x 1 = x 2).
Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. Minkowski's question mark function : Derivatives vanish on the rationals. Weierstrass function : is an example of continuous function that is nowhere differentiable
One application is the definition of inverse trigonometric functions. For example, the cosine function is injective when restricted to the interval [0, π]. The image of this restriction is the interval [−1, 1], and thus the restriction has an inverse function from [−1, 1] to [0, π], which is called arccosine and is denoted arccos.
Nowhere continuous function: is not continuous at any point of its domain; for example, the Dirichlet function. Homeomorphism: is a bijective function that is also continuous, and whose inverse is continuous. Open function: maps open sets to open sets. Closed function: maps closed sets to closed sets.
A common example of a sigmoid function is the logistic function, which is defined by the formula: [1] ... Sign function – Mathematical function returning -1, 0 or 1;
In mathematics, an elementary function is a function of a single variable (typically real or complex) that is defined as taking sums, products, roots and compositions of finitely many polynomial, rational, trigonometric, hyperbolic, and exponential functions, and their inverses (e.g., arcsin, log, or x 1/n).
The easiest example of a completely multiplicative function is a monomial with leading coefficient 1: For any particular positive integer n, define f(a) = a n. Then f ( bc ) = ( bc ) n = b n c n = f ( b ) f ( c ), and f (1) = 1 n = 1.
A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain. [1] Well-known examples of convex functions of a single variable include a linear function = (where is a real number), a quadratic function (as a nonnegative real number) and an exponential function (as a ...