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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
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.
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.
The given examples are real functions, to illustrate the symmetry of their graphs. Even functions = is an example of an even function. A real function f is even if ...
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 function () = has ″ = >, so f is a convex function. It is also strongly convex (and hence strictly convex too), with strong convexity constant 2. The function () = has ″ =, so f is a convex function. It is strictly convex, even though the second derivative is not strictly positive at all points.
For example, simple functions attain only a finite number of values. Some authors also require simple functions to be measurable, as used in practice. A basic example of a simple function is the floor function over the half-open interval [1, 9), whose only values are {1
A function is bijective if it is both injective and surjective. A bijective function is also called a bijection or a one-to-one correspondence (not to be confused with one-to-one function, which refers to injection). A function is bijective if and only if every possible image is mapped to by exactly one argument. [1]