* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

Every character is automatically continuous from A to C, since the kernel of a character is a maximal ideal, which is closed.

* Every continuous map from a compact space to a Hausdorff space is closed and proper ( i. e., the pre-image of a compact set is compact.

* Pseudocompact: Every real-valued continuous function on the space is bounded.

Every contraction mapping is Lipschitz continuous and hence uniformly continuous ( for a Lipschitz continuous function, the constant k is no longer necessarily less than 1 ).

Every continuous map f: X → Y induces an algebra homomorphism C ( f ): C ( Y ) → C ( X ) by the rule C ( f )( φ ) = φ o f for every φ in C ( Y ).

Every space filling curve hits some points multiple times, and does not have a continuous inverse.

* Every separable metric space is isometric to a subset of C (), the separable Banach space of continuous functions → R, with the supremum norm.

Every uniformly continuous function between metric spaces is continuous.

Every continuous function on a compact set is uniformly continuous.

Every topological group can be viewed as a uniform space in two ways ; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.

Every continuous function in the function space can be represented as a linear combination of basis functions, just as every vector in a vector space can be represented as a linear combination of basis vectors.

* Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous.

Every embedding is injective and continuous.

Every map that is injective, continuous and either open or closed is an embedding ; however there are also embeddings which are neither open nor closed.

* Every compact Hausdorff space of weight at most ( see Aleph number ) is the continuous image of ( this does not need the continuum hypothesis, but is less interesting in its absence ).

Every place south of the Antarctic Circle experiences a period of twenty-four hours ' continuous daylight at least once per year, and a period of twenty-four hours ' continuous night time at least once per year.

: Every set has a choice function.

Every such subset has a smallest element, so to specify our choice function we can simply say that it maps each set to the least element of that set.

** Every surjective function has a right inverse.

: Every effectively calculable function is a computable function.

Every effectively calculable function ( effectively decidable predicate ) is general recursive italics

Every effectively calculable function ( effectively decidable predicate ) is general recursive.

Every bijective function g has an inverse g < sup >− 1 </ sup >, such that gg < sup >− 1 </ sup > = I ;

Every entire function can be represented as a power series that converges uniformly on compact sets.

Every holomorphic function can be separated into its real and imaginary parts, and each of these is a solution of Laplace's equation on R < sup > 2 </ sup >.

Every holomorphic function is analytic.

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

Every polynomial P in x corresponds to a function, ƒ ( x )

Every primitive recursive function is a general recursive function.

Every time another object or customer enters the line to wait, they join the end of the line and represent the “ enqueue ” function.

Every function is a method and methods are always called on an object.

Every type that is a member of the type class defines a function that will extract the data from the string representation of the dumped data.

Every output of an encoder can be described by its own transfer function, which is closely related to the generator polynomial.

Every homomorphism f: G → H of Lie groups induces a homomorphism between the corresponding Lie algebras and.

Every locally constant function from the real numbers R to R is constant by the connectedness of R. But the function f from the rationals Q to R, defined by f ( x ) = 0 for x < π, and f ( x ) = 1 for x > π, is locally constant ( here we use the fact that π is irrational and that therefore the two sets

Every empty function is constant, vacuously, since there are no x and y in A for which f ( x ) and f ( y ) are different when A is the empty set.

Theorem Every self-adjoint f in A * can be written as f

Every order-preserving self-map f of a cpo ( P, ⊥) has a least fixpoint.

Every bounded positive-definite measure μ on G satisfies μ ( 1 ) ≥ 0. improved this criterion by showing that it is sufficient to ask that, for every continuous positive-definite compactly supported function f on G, the function Δ < sup >– ½ </ sup > f has non-negative integral with respect to Haar measure, where Δ denotes the modular function.

Every morphism f: G → H in Grp has a category-theoretic kernel ( given by the ordinary kernel of algebra ker f =

Every isogeny f: A → B is automatically a group homomorphism between the groups of k-valued points of A and B, for any field k over which f is defined.

Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary.

Every deterministic complexity class ( DSPACE ( f ( n )), DTIME ( f ( n )) for all f ( n )) is closed under complement, because one can simply add a last step to the algorithm which reverses the answer.

* Every point x of X is isolated in its fiber f < sup >− 1 </ sup >( f ( x )).