This exercise provides an example of a function f: R² → R such that all first partial derivatives exist everywhere, but ƒ is discontinuous. Let f(x, y) = { 2xy² x²+y4 if (x, y) = (0,0) (0,0) if (x, y) = (0,0). (a) Calculate (D₁f)(x,y) and (D₂f)(x, y) at all points (x,y) different from (0,0). (b) Calculate (D₁ƒ)(0,0) and (D₂f)(0,0) directly from the definition of derivative. (c) Show that f is discontinuous at (0,0) by showing limy→o ƒ(y², y) ‡ 0.

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