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If nonempty f: X → Y is injective, construct a left inverse g: Y → X as follows: for all y ∈ Y, if y is in the image of f, then there exists x ∈ X such that f(x) = y. Let g(y) = x; this definition is unique because f is injective. Otherwise, let g(y) be an arbitrary element of X. For all x ∈ X, f(x) is in the image of f.
The set of all functions f: X → X is called the full transformation semigroup [6] or symmetric semigroup [7] on X. (One can actually define two semigroups depending how one defines the semigroup operation as the left or right composition of functions. [8]) Composition of a shear mapping (red) and a clockwise rotation by 45° (green). On the ...
Cauchy's functional equation is the functional equation: (+) = + (). A function that solves this equation is called an additive function.Over the rational numbers, it can be shown using elementary algebra that there is a single family of solutions, namely : for any rational constant .
The Caputo fractional derivative is motivated from the Riemann–Liouville fractional integral.Let be continuous on (,), then the Riemann–Liouville fractional integral states that
Middle Riemann sum of x ↦ x 3 over [0, 2] using 4 subintervals. For the midpoint rule, the function is approximated by its values at the midpoints of the subintervals. This gives f(a + Δx/2) for the first subinterval, f(a + 3Δx/2) for the next one, and so on until f(b − Δx/2).
The notation convention chosen here (with W 0 and W −1) follows the canonical reference on the Lambert W function by Corless, Gonnet, Hare, Jeffrey and Knuth. [3]The name "product logarithm" can be understood as follows: since the inverse function of f(w) = e w is termed the logarithm, it makes sense to call the inverse "function" of the product we w the "product logarithm".
The Chebyshev series converges to f(x) if the function is piecewise smooth and continuous. The smoothness requirement can be relaxed in most cases – as long as there are a finite number of discontinuities in f(x) and its derivatives. At a discontinuity, the series will converge to the average of the right and left limits.
In category theory, a branch of mathematics, the formal criteria for adjoint functors are criteria for the existence of a left or right adjoint of a given functor.. One criterion is the following, which first appeared in Peter J. Freyd's 1964 book Abelian Categories, [1] an Introduction to the Theory of Functors: