Question

Differential Equations

If E X and if f is a function defined on X, the restriction of f to E is the functiong whose domain of definition is E, such that g(p) =f(p) for p E. Define f and gon R by: f(0, 0) = g(0, 0) = 0, f(x, y) = xy2/(x² + y4), g(x, y) = xy²/(x3+y4)if (x, y) is not equal (0, 0). Prove that f is bounded on R2, 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 R2 are continuous!


Answer

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