Question

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is somewhat difficult. In practice, it is often easier to show a stronger condition: if each partial derivative OJi = 1, ... , n, is continuous in a disc around p = _ (a1.... , an), then f is differentiable дх, (a1,., an) Put differently: if f is continuously differentiable at p, it is differentiable at However, just as in the one-variable case, there are functions that are differentiable but not at p = p. continuously differentiable. Consider the function f : R2 - R ya2 sin(z 0, x 0 f(x, y) т— 0 0, f is not continuous at x = 0, in that lim fa(a, y) doesn't exist (a) Show that if y a+0 (b) Nevertheless, show that f is differentiable when x with f(0, ) 0. To be precise, show that 0 (this is clearly the only problematic case), |f(h, y lim 2 k) f(0, y) 0 Vh2k2 _ - = 0 (h,k)-(0,0)

Answer 1

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