Could you explain how to show 7.5 in detail? Definition. Let X and Y be topological spaces. A function or map f : X → Y is acontinuous function or continuous map if and only if for every open set U in Y,f-'(U) is open in X.Definition. Let f : X → Y be a function between topological spaces X and Y, andlet x e X. Then f is continuous at the point x if and only if for every open setV containing f(x), there is an open set U containing x such that f(U) c V. Thus afunction f : X → Y is continuous if and only if it is continuous at each point.Theorem 7.5. A function f : Rstdx in R and ɛ > 0, there is a 8 > 0 such that for every y E R with d(x, y) < 8, thend(f(x), f(y)) < ɛ.→ Rstd is continuous if and only if for every point

Answered: Image /qna-images/answer/67705e34-3e95-4885-8f3e-177ff041f974.jpg

Answered: Theorem 7.5. A function f : Rstd Rstd…

Theorem 7.5. A function f : Rstd Rstd is continuous if and only if for every point x in R and ɛ > 0, there is a 8 > 0 such that for every y E R with d(x, y) < 8, then d(f(x), f(y)) < ɛ.

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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Could you explain how to show 7.5 in detail?

Definition. Let X and Y be topological spaces. A function or map f : X → Y is a
continuous function or continuous map if and only if for every open set U in Y,
f-'(U) is open in X.
Definition. Let f : X → Y be a function between topological spaces X and Y, and
let x e X. Then f is continuous at the point x if and only if for every open set
V containing f(x), there is an open set U containing x such that f(U) c V. Thus a
function f : X → Y is continuous if and only if it is continuous at each point.
Theorem 7.5. A function f : Rstd
x in R and ɛ > 0, there is a 8 > 0 such that for every y E R with d(x, y) < 8, then
d(f(x), f(y)) < ɛ.
→ Rstd is continuous if and only if for every point

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