关于复合函数的连续性与间断点类型的探讨

Discussion on Continuity and Discontinuous Points of Composite Functions

摘要: 设函数y=g(x)，y₀=g(x₀)，z=f(y)，以及邻域U(x₀)⊂D_g，U(y₀)⊂D_f . 根据[1]中的论述, 如果(i)g(x)在x0点连续，(ii)f(y)在y₀点连续，则复合函数f[g(x)] 在x₀点连续。本文将通过构造简单的实例论证当(i)、(ii)中至少有一条不成立时，f[g(x)] 在x₀点不一定能保证连续性，且当f[g(x)] 在x₀点间断时，其间断点类型也未必与x₀之于g(x)及y₀之于f(y)的情况相同。

Abstract: Suppose y = g(x), y₀ = g(x₀), z = f (y), U (x₀) ⊂ D_g , U (y₀) ⊂ D_f , by [1], we have following conclusions. If (i) g(x) is continuous at x₀, and (ii) f (y) is continuous at y₀, then com- posite function f [g(x)] is continuous at x0. In this paper, we will use some examples to illustrate that f [g(x)] is not necessarily continuous at x₀ under the assumption of (i) and (ii) at least one of them is not established. Moreover, when f [g(x)] is discontinuous at x₀, the types of discontinuous points are not necessary to maintain the same types of x₀ for g(x) and y₀ for f (y).