Existence of one-sided limits for a non-decreasing function

问题

Let be non-decreasing and is arbitrary. How to show that both of and exist?

回答

Hint: For the limit from the left, let be the supremum of the over all . So you need first to show that the set of all as ranges over is bounded. The existence of the supremum follows immediately, and it is not hard to check, from the definition of supremum, that is the required limit from the left. The limit from the right is dealt with analogously.

参考

real analysis - How to prove the existence of one-sided limits for a non-decreasing function - Mathematics Stack Exchange

进一步的问题

Theorem: Let be a differentiable function and suppose that for every , then there exists .

此时极限可能是无穷，又该怎么证明呢？

此时可以考虑利用已经证明的结论，在端点 0 的右边每一点处都有右极限，考虑这些右极限的下确界。

参考资料

• 问题实际上来自 Rudin-Real analysis-P95-Theorem 4.29