论文部分内容阅读
1963年6月数学通报上刊登了顏怀曾同志所写的“周期函数的最小正周期”一文其中証明了下述定理: 定理。任一非常值的連續周期函数f(x)必有最小正周期。这一定理和1959年12月数学通报吳品三同志在“几篇有缺点的文章”中証明过的定理“設f(x)是連續的周期函数除f(x)=c外f(x)均有最小正周期存在”是一样的。由狄里克萊函数的例子知道,有些不連續周期函数是沒有最小正周期的。但也有不連續周期函数具有最小正周期的,如f(x)=tgx就是最簡单的例子。π是它的最小正周期,x=(2K+1)π/2(K=0,±1,±2,…)是一些不連續点。于是发生下面的問題: 哪些不連續周期函数有最小正周期呢?前述两篇文章并未提及。但綜合两篇文章証法的精神可进一步推出下面一个定理:
In June 1963, the Mathematics Bulletin published the article “Minimum Positive Period of Periodic Function” written by Yan Huai-Zeng, which proved the following theorem: Theorem. Any non-negative continuous-period function f(x) must have the minimum positive period. This theorem and the 1959 December mathematics report Prof. Wu Pinsan’s theorem proved in “Several Disadvantaged Articles”: Let f(x) be a continuous periodic function except f(x)=c. f(x) “There is a minimum positive cycle exists” is the same. From the example of the Dirichlet function, it is known that some discrete periodic functions do not have a minimum positive period. However, there are discontinuity periodic functions with the smallest positive period, such as f(x) = tgx is the simplest example. π is its smallest positive period, and x=(2K+1)π/2 (K=0,±1,±2,...) are some discontinuities. The following question then occurs: Which discontinuity periodic function has the smallest positive period? The two previous articles have not mentioned it. But synthesizing the spirit of the two article proofs can further introduce the following theorem: