The "trouble spot" for f is the origin, so for it would be when g passes through the origin. Using the chain rule, what would be the expression for the derivative of at the trouble spot?
let f : R^2 -> R be defined as:
f = x^3 / (x^2 + y^2) if (x,y) =/= 0
and 0 if (x,y) = 0.
let g be any curve in R^2 that passes through the origin. then show that f o g is differentiable while f itself is not. so i have that g: R -> R^2 and using the definition of the limit i have lim (h->0) [f(g(h)) - f(g(0)) - mh]/h which we want to show should = 0. i am evaluating the derivative at 0 because at any other point where (x,y) =/= 0 then f is differentiable so the only trouble spot comes from (x,y) = 0. however i am stuck since i don't know anything about g, just that its a curve that passes through the origin which does not help me figure out what f(g(h)) could be. any help is greatly appreciated. thanks.
so i'm assuming g: R -> R^2 is given by g(t) = (u(t), v(t)) so D(fog) = f_u * u'(t) + f_v * v'(t). so at 0 it would be m = D(fog(0)) = f_u(0) * u'(0) + f_v(0) * v'(0). i am not quite sure what to do next. i have the partial of f with respect to u which i don't know how to evaluate. trying to use the limit definition, f(g(h)) would become u(h)^3 / ((u(h)^2 + v(h)^2) which doesn't look promising. i know that at least one of u(t) or v(t) is not constant because if they both were, it would just be a point. so the quotient should be some function of h but there is no telling what kind of function of h it is and whether or not it will allow the limit to be 0. is the limit definition the way to go with this problem?