Search results
Results from the WOW.Com Content Network
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.
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.
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 complex valued function is conjugate symmetric if and only if its real part is an even function and its imaginary part is an odd function. A typical example of a conjugate symmetric function is the cis function = + Conjugate antisymmetry:
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).
Typical examples of functions that are not analytic are The absolute value function when defined on the set of real numbers or complex numbers is not everywhere analytic because it is not differentiable at 0. Piecewise defined functions (functions given by different formulae in different regions) are typically not analytic where the pieces meet.
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.