Consider the function f : R2 - R defined by f (x, y) = ry 1 2 ty (x, y) # (0, 0) (x, y) = (0,0) (a) (6 points) Calculate fr and fy, and show that one of them is continuous (both are continuous, but just check one). Is f is differentiable everywhere? Hint: be careful calculating fx(0, 0) and f, (0, 0). Use polar coordinates to check continuity at 0. (b) (2 points) Find the contour for f corresponding to c = 0, that is, find the set of points (x, y) such that f(x, y) =0. Sketch this contour. (c) (1 point) The contour from part (b) splits R2 into different regions. What is the sign (positive or negative) of f on each region? Hint: since f is differentiable (and hence continuous) by part (a) it will be either positive or negative on the whole of each region, and so you can just pick a random point (a, b) in each region and check the sign of f(a, b). (d) (2 points) Calculate the four mixed partial derivatives of f at (0, 0). Conclude that fry and fyr can't both be continuous everywhere