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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.

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
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
Transcribed Image Text: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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