2023高三·全国·专题练习
1 . 已知
是定义在
上的函数,如果存在常数
,对区间
的任意划分:
,
恒成立,则称函数
为区间
上的“有界变差函数”;
(1)试判断函数
是否为区间
上的“有界变差函数”,若是,求出M的最小值;若不是,说明理由;
(2)若
与
均为区间
上的“有界变差函数”,证明:
是区间
上的“有界变差函数”;
(3)证明:函数
不是
上的“有界变差函数”.
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f5f4dffc65e0fc5d24367a9d4e5c997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/2480f87a11c4cd450bc9454ea7276722.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/ffce5e1754afb41ac580bd2b3b8638a1.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/36d20a32df93387be6b6c1e296d3c867.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/8f5f4dffc65e0fc5d24367a9d4e5c997.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
(1)试判断函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/dff8704285d8c14ae2bd82f9196501c7.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/44d51992c05a557cf6058664f1f8961e.png)
(2)若
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/6a1cfb60420ff7e72c1b9d64f69ae063.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/e915b67f8f747698b8b46d37bc453667.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/b148ebfd8746a83018c9bfd0314eb938.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/627565d32e529cafcd2744d006ec6de2.png)
(3)证明:函数
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/0aad34eed5353a44b87abbf959055a53.png)
![](https://staticzujuan.xkw.com/quesimg/Upload/formula/304226ca50149b49702928e44d565964.png)
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