Question

derivative at p Bonus) It turns out that showing a function of multiple variables f(11, 12,...,In) is differentiable is somewhat difficult. In practice, it is often easier to show a stronger condition: if each partial af ax;' i = 1,..., n, is continuous in a disc around p = (as....,an), then is differentiable (21,...,0m). Put differently: if f is continuously differentiable at p, it is differentiable at p. However, just as in the one-variable case, there are functions that are differentiable but not continuously differentiable. Consider the function f :R? +R Syaº sin(x-), **0 f(x,y) X=0 (a) Show that if y0, S. is not continuous at : = 0, in that lim fr(a,y) doesn't exist. (b) Nevertheless, show that f is differentiable when I = 0 (this is clearly the only problematic case), with f'(0,y) = 0. To be precise, show that f( hy+k) - f(0,y) - lim = 0 + (0.5) - 01 -

Answer 1

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