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Rudin Ch 4, p. 99 #7. If E ⊂ X and if f is a function...

Rudin Ch 4, p. 99 #7. If E ⊂ X and if f is a function defined on X, the restriction of f to E is the function g whose domain of definition is E, such that g(p) = f (p) for p ∈ E. Define f and g on 2 by: f (0, 0) = g(0, 0) = 0, f (x, y) = € xy 2 x 2 + y 4 , g(x, y) = € xy 2 x 2 + y 6 if (x, y) ≠ (0, 0). Prove that f is bounded on 2 , that g is unbounded in every neighborhood of (0, 0) and that f is not continuous at (0, 0); nevertheless, the restrictions of both f and g to every straight line in 2 are continuous! .Explain step by step in detal

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Colby Messinger answered 2 years ago
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