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Let P be the Cantor set constructed in Sec. 2.44. Let f be a bounded real function on [0,1] which is continuous at every point outside P. Prove that f ∈ℛ on [0,1]. Hint: P can be covered by finitely many segments whose total length can be made as small as desired. Proceed as in Theorem 6.10 . | Numerade

Let P be the Cantor set constructed in Sec. 2.44. Let f be a bounded real function on [0,1] which is continuous at every point outside P. Prove that f ∈ℛ on [0,1]. Hint: P can be covered by finitely many segments whose total length can be made as small as desired. Proceed as in Theorem 6.10 . | Numerade

2021年6月8日

$SOLVED:Let M be the space of all bounded real functions defined on the interval [a, b], with metric \rho(f, g)=\sup |f-g|. By the length of the curve y=f(x) \quad(a \leqslant x \leqslant b) is meant the functional L(f)=\sup \sum_{i=1}^{n} \sqrt{\left(x_{i}-x_{i-1}\right)^{2} \left(f\left(x_{i}\right)-f\left(x_{i-1}\right)\right)^{2}} where the least upper bound (which may equal \infty) is taken over all possible partitions of [a, b] obtained by introducing points of subdivision x, x, , x_{2}, \$

SOLVED:Let M be the space of all bounded real functions defined on the interval [a, b], with metric \rho(f, g)=\sup |f-g|. By the length of the curve y=f(x) \quad(a \leqslant x \leqslant b) is meant the functional L(f)=\sup \sum_{i=1}^{n} \sqrt{\left(x_{i}-x_{i-1}\right)^{2} \left(f\left(x_{i}\right)-f\left(x_{i-1}\right)\right)^{2}} where the least upper bound (which may equal \infty) is taken over all possible partitions of [a, b] obtained by introducing points of subdivision x, x, , x_{2}, \

Bounded Analytic Functions

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