“continuous function”的概念、定义、习题解题方法及翻译-数学辞典

单词

continuous function

释义

continuous function

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 xn converges to a, f(xn) 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 dN(f(m),f(a))<εwhenever dM(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.

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