if and only if
inf｢sup(f(x))Ƨ[a,b]｣
= sup｢inf(f(x))Ƨ[a,b]｣
.
The common value is denoted by f(x)ᐃx｢a,b｣.ᐥ
┄
inf{U:U=｢sup{f(x):x⸤i−1⸥≤x≤x⸤i⸥}·(x⸤i⸥−x⸤i−1⸥) Σi=0,n｣}
= sup{L:L=｢inf{f(x):x⸤i−1⸥≤x≤x⸤i⸥}·(x⸤i⸥−x⸤i−1⸥) Σi=0,n｣}
┄
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7103 Riemann integrable |
| by EngliSea on 2020-07-16 | |
| ᐥA function f defined and bounded on [a,b] is Riemann integrable on [a,b] if and only if inf｢sup(f(x))Ƨ[a,b]｣ = sup｢inf(f(x))Ƨ[a,b]｣ . The common value is denoted by f(x)ᐃx｢a,b｣.ᐥ ┄ inf{U:U=｢sup{f(x):x⸤i−1⸥≤x≤x⸤i⸥}·(x⸤i⸥−x⸤i−1⸥) Σi=0,n｣} = sup{L:L=｢inf{f(x):x⸤i−1⸥≤x≤x⸤i⸥}·(x⸤i⸥−x⸤i−1⸥) Σi=0,n｣} ┄ |
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