A step function f on [a, b] is a function for which there are a finite number of disjointintervals Ib ... , I. with [a, b] = I 1 u · · ·ui. for which f is constant on each of theintervals. (b) Let f be a continuous (and thus uniformly continuous) function on [a, b]. Showthat, given ɛ > 0, there is a step function g, defined on [a, b] such thatIg.(x) – f(x)| <ɛ for every x e [a, b]and< E(b – a).

Answered: Image /qna-images/answer/c40b72f3-88d6-4ca0-85f9-ffad363c7b83.jpg

) Let f be a continuous (and thus uniformly continuous) function on [a, b]. Show that, given e > 0, there is a step function g, defined on [a, b] such that Ig:(x) – f(x)| < ɛ for every x e [a, b] and ge f < e(b – a).

) Let f be a continuous (and thus uniformly continuous) function on [a, b]. Show that, given e > 0, there is a step function g, defined on [a, b] such that Ig:(x) – f(x)| < ɛ for every x e [a, b] and ge f < e(b – a).

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

A step function f on [a, b] is a function for which there are a finite number of disjoint
intervals Ib ... , I. with [a, b] = I 1 u · · ·ui. for which f is constant on each of the
intervals.

(b) Let f be a continuous (and thus uniformly continuous) function on [a, b]. Show
that, given ɛ > 0, there is a step function g, defined on [a, b] such that
Ig.(x) – f(x)| <ɛ for every x e [a, b]
and
< E(b – a).

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