Suppose the function g is on the interval [a, b], and [g(x)|


Suppose the function g is on the interval [a, b], and [g(x)|<B for all x E R.<br>I) Prove that g(x)² – g(y)²|< 2B|g(x) – g(y)| for all x, y e [a, b].<br>-<br>II) Prove that if a partition P determined the closed interval [x;, x¡+1], then<br>M;(g²) – m;(g²)< 2B(M;(g) – m;(g)),<br>where M;(*) and m; (*) are the supremum and infimum of the function * on [x;, xi+1].<br>III) Prove that for any partition P of [a, b], U(g², P) – L(g², P) < 2B(U(g, P) – L(g, P)).<br>IV) Prove that if the function g is integrable on [a, b], then so is g?.<br>

Extracted text: Suppose the function g is on the interval [a, b], and [g(x)|
Jun 05, 2022
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