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.

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Description: Could you explain how to show 9.19 in detail? Theorem 9.19. A function f from a metric space (X, dx) to a metric space (Y, dy) iscontinuous at the point x (in the topological sense) if and only if for every ɛ > 0 thereexists a 8 > 0 such that for eve

Elements Of Modern Algebra

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Author:Gilbert, Linda, Jimmie

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Chapter6: More On Rings

Section6.2: Ring Homomorphisms

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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, 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.

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