集上的连续函数及一致连续函数的延拓问题

已知函数在开区间内一致连续,可证得在处有有限极限(指单侧极限存在)。因此,如果将极限值分别作为在处的值,则可以被延拓到闭区间,且在上一致连续。同样,把连续及一致连续的概念推广到一般的集合上,也有类似的结论。

The Continuation Problem of Continuous Function and Uniformly Continuous Function in Sets

If f(x) is uniformly continuous function in the open interval(a,b),the existence of limit of f(x) on the point of a and b can be proved(Here it refers to the existence of single-sided limit).Consequently,if limited value is regarded as the value of f(x) on the point of a and b,f(x) will be prolonged to closed interval ,and will be uniformly continuous in .Similarly,if the concepts of continuity and uniform continuity were extended to a general set ER ,the same conclusion could be drawn.