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Page "Singular function" ¶ 7
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If and ƒ
If ƒ is differentiable, this is equivalent to:
If is continuous in an open set Ω and the partial derivatives of ƒ with respect to x and y exist in Ω, and satisfies the Cauchy – Riemann equations throughout Ω, then ƒ is holomorphic ( and thus analytic ).
* If ƒ ( z ) is locally integrable in an open domain Ω ⊂ C, and satisfies the Cauchy – Riemann equations weakly, then ƒ agrees almost everywhere with an analytic function in Ω.
If X is a set and M is a complete metric space, then the set B ( X, M ) of all bounded functions ƒ from X to M is a complete metric space.
If X is a topological space and M is a complete metric space, then the set C < sub > b </ sub >( X, M ) consisting of all continuous bounded functions ƒ from X to M is a closed subspace of B ( X, M ) and hence also complete.
If and are groups, a homomorphism from to is a function ƒ: → such that
If the limit exists, we say that ƒ is complex-differentiable at the point z < sub > 0 </ sub >.
If ƒ is complex differentiable at every point z < sub > 0 </ sub > in an open set U, we say that ƒ is holomorphic on U. We say that ƒ is holomorphic at the point z < sub > 0 </ sub > if it is holomorphic on some neighborhood of z < sub > 0 </ sub >.
In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa.
If ƒ maps X to Y, then ƒ < sup >– 1 </ sup > maps Y back to X.
If ƒ is invertible, the function g is unique ; in other words, there can be at most one function g satisfying this property.
If the domain is the real numbers, then each element in Y would correspond to two different elements in X (± x ), and therefore ƒ would not be invertible.
If ƒ is an invertible function with domain X and range Y, then
If an inverse function exists for a given function ƒ, it is unique: it must be the inverse relation.
If ƒ is real-valued, the polynomial function can be taken over R.
If D is a derivation at x, then D ( ƒ )
If ƒ and g are elements of a C *- algebra, f * and g * denote their respective adjoints.
If ƒ: A → B is a homomorphism between two algebraic structures ( such as homomorphism of groups, or a linear map between vector spaces ), then the relation ≡ defined by
* Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ < sub > n </ sub > converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions.

If and x
* If it is required to use a single number X as an estimate for the value of numbers, then the arithmetic mean does this best, in the sense of minimizing the sum of squares ( x < sub > i </ sub > − X )< sup > 2 </ sup > of the residuals.
If F is algebraically closed and p ( x ) is an irreducible polynomial of F, then it has some root a and therefore p ( x ) is a multiple of x − a.
If P is a program which outputs a string x, then P is a description of x.
If M is a Turing Machine which, on input w, outputs string x, then the concatenated string < M > w is a description of x.
If F is an antiderivative of f, and the function f is defined on some interval, then every other antiderivative G of f differs from F by a constant: there exists a number C such that G ( x ) = F ( x ) + C for all x.
If the filter shows amplitude ripple within the passband, the x dB point refers to the point where the gain is x dB below the nominal passband gain rather than x dB below the maximum gain.
* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.
* If G is a locally compact Hausdorff topological group and μ its Haar measure, then the Banach space L < sup > 1 </ sup >( G ) of all μ-integrable functions on G becomes a Banach algebra under the convolution xy ( g ) = ∫ x ( h ) y ( h < sup >− 1 </ sup > g )( h ) for x, y in L < sup > 1 </ sup >( G ).
If a Banach algebra has unit 1, then 1 cannot be a commutator ; i. e., for any x, y ∈ A.
If x is held fixed, then the Bessel functions are entire functions of α.
If the exponent r is even, then the inequality is valid for all real numbers x.
If x is a member of A, then it is also said that x belongs to A, or that x is in A.
If ( x < sub > 1 </ sub >, x < sub > 2 </ sub >, x < sub > 3 </ sub >) are the Cartesian coordinates and ( u < sub > 1 </ sub >, u < sub > 2 </ sub >, u < sub > 3 </ sub >) are the orthogonal coordinates, then

ƒ and x
The function ƒ ( x ) may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote.
has a limit of +∞ as, ƒ ( x ) has the vertical asymptote, even though ƒ ( 0 ) = 5.
The horizontal line y = c is a horizontal asymptote of the function y = ƒ ( x ) if
In the first case, ƒ ( x ) has y = c as asymptote when x tends to −∞, and in the second that ƒ ( x ) has y = c as an asymptote as x tends to +∞
In the first case the line is an oblique asymptote of ƒ ( x ) when x tends to +∞, and in the second case the line is an oblique asymptote of ƒ ( x ) when x tends to −∞
An example is ƒ ( x ) = x − 1 / x, which has the oblique asymptote y = x ( m = 1, n = 0 ) as seen in the limits

0.693 seconds.