“continuous function”的概念、定义、翻译、参考文献-科学参考

continuous function

Physics

A function f(x) is continuous at x=a if the limit of f(x) as x approaches a is f(a). A function that does not satisfy this condition is said to be a discontinuous function.

Mathematics

The function f:S → ℝ is continuous at a in S if f(x) → f(a) as x → a (see limit). Rigorously, this means that, given any ε‎>0, there exists δ‎>0 such that |f(x)−f(a)| < ε‎ whenever |x−a|< δ‎ and x ∈ S. This is equivalent to saying that whenever a sequence x_n converges to a, f(x_n) converges to f(a). The function f is continuous on S if it is continuous at each point of S. The following algebra of limits holds:
1. (i) The sum of two continuous functions is continuous.
2. (ii) The product of two continuous functions is continuous.
3. (iii) The quotient of two continuous functions is continuous provided the denominator is not zero.
4. (iv) Suppose that f is continuous at a, that f(a) = b and that g is continuous at b. Then the composition g∘f is continuous at a.
5. (v) Any differentiable function is continuous. The converse is not true as the modulus function is continuous but not differentiable.
See blancmange function, intermediate value theorem, Weierstrass’ theorem.

This notion of continuity generalizes readily to metric spaces: a function f:M → N between metric spaces is continuous at a ∈ M if, given any ∈>0, there exists δ‎>0 such that d_N(f(m),f(a))<ε‎whenever d_M(m,a)|<δ‎ and m ∈ M. This further generalizes to topological spaces: a function f:M → N between topological spaces is continuous at aε‎M if, given any open subset (see open set) U containing f(a), the pre-image f^–1(U) is open in M.

Computer

A function from one partially ordered set to another having the property, roughly speaking, that least upper bounds are preserved. A function
$f : S \to T$
is said to be continuous if, for every directed subset X of S, f maps the least upper bound of X to the least upper bound of the image of X under f. Continuous functions are significant in denotational semantics since they correspond to the requirement that a computational process produces arbitrarily close approximations to the final output, given arbitrarily close approximations to the total input.
A continuous function f(x) has no breaks or instantaneous changes in value. In the hierarchy of mathematical functions the smoothest are those, such as sin x, cos x, that can be differentiated any number of times, always producing a continuous function.

单词	continuous function
释义	continuous function Physics A function f(x) is continuous at x=a if the limit of f(x) as x approaches a is f(a). A function that does not satisfy this condition is said to be a discontinuous function. Mathematics The function f:S → ℝ is continuous at a in S if f(x) → f(a) as x → a (see limit). Rigorously, this means that, given any ε‎>0, there exists δ‎>0 such that \|f(x)−f(a)\| < ε‎ whenever \|x−a\|< δ‎ and x ∈ S. This is equivalent to saying that whenever a sequence x_n converges to a, f(x_n) converges to f(a). The function f is continuous on S if it is continuous at each point of S. The following algebra of limits holds: (i) The sum of two continuous functions is continuous. (ii) The product of two continuous functions is continuous. (iii) The quotient of two continuous functions is continuous provided the denominator is not zero. (iv) Suppose that f is continuous at a, that f(a) = b and that g is continuous at b. Then the composition g∘f is continuous at a. (v) Any differentiable function is continuous. The converse is not true as the modulus function is continuous but not differentiable. See blancmange function, intermediate value theorem, Weierstrass’ theorem. This notion of continuity generalizes readily to metric spaces: a function f:M → N between metric spaces is continuous at a ∈ M if, given any ∈>0, there exists δ‎>0 such that d_N(f(m),f(a))<ε‎whenever d_M(m,a)\|<δ‎ and m ∈ M. This further generalizes to topological spaces: a function f:M → N between topological spaces is continuous at aε‎M if, given any open subset (see open set) U containing f(a), the pre-image f^–1(U) is open in M. Computer A function from one partially ordered set to another having the property, roughly speaking, that least upper bounds are preserved. A function $f : S \to T$ is said to be continuous if, for every directed subset X of S, f maps the least upper bound of X to the least upper bound of the image of X under f. Continuous functions are significant in denotational semantics since they correspond to the requirement that a computational process produces arbitrarily close approximations to the final output, given arbitrarily close approximations to the total input. A continuous function f(x) has no breaks or instantaneous changes in value. In the hierarchy of mathematical functions the smoothest are those, such as sin x, cos x, that can be differentiated any number of times, always producing a continuous function.
随便看	differential winding differentiation differentiation index differentiation under the integral sign differentiator Diffie Hellman key agreement diffraction diffraction grating diffraction limit diffraction of atoms diffraction pattern diffraction rings diffuse coevolution diffused junction diffuse interstellar bands diffuse nebula diffuse pollution diffuser diffuse reflection diffusion diffusion address diffusion cloud chamber diffusion coefficient diffusion constant diffusion-controlled growth