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Theorem 9.19. A function f from a metric space (X, dx) to a metric space (Y, dy) is continuous at the point x (in the topological sense) if and only if for every ɛ > 0 there exists a 8 > 0 such that for every y e X, if dx(x, y) < 8, then dy(f(x), f(y)) < ɛ. The function f is continuous if and only if it is continuous at every point x E X.

Theorem 9.19. A function f from a metric space (X, dx) to a metric space (Y, dy) is continuous at the point x (in the topological sense) if and only if for every ɛ > 0 there exists a 8 > 0 such that for every y e X, if dx(x, y) < 8, then dy(f(x), f(y)) < ɛ. The function f is continuous if and only if it is continuous at every point x E X.

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 9.19 in detail?

**Theorem 9.19.** A function \( f \) from a metric space \((X, d_X)\) to a metric space \((Y, d_Y)\) is continuous at the point \( x \) (in the topological sense) if and only if for every \( \varepsilon > 0 \) there exists a \( \delta > 0 \) such that for every \( y \in X \), if \( d_X(x, y) < \delta \), then \( d_Y(f(x), f(y)) < \varepsilon \). The function \( f \) is continuous if and only if it is continuous at every point \( x \in X \).
Transcribed Image Text:**Theorem 9.19.** A function \( f \) from a metric space \((X, d_X)\) to a metric space \((Y, d_Y)\) is continuous at the point \( x \) (in the topological sense) if and only if for every \( \varepsilon > 0 \) there exists a \( \delta > 0 \) such that for every \( y \in X \), if \( d_X(x, y) < \delta \), then \( d_Y(f(x), f(y)) < \varepsilon \). The function \( f \) is continuous if and only if it is continuous at every point \( x \in X \).
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