Let a function be given by
$f(0,0) = 0$ and
$f(x,y) = \frac {sin(x^3+y^3)}{x^2+y^2}$ for $(x,y)\neq (0,0)$.
I have already proved that all directional derivatives exist.
Is the function differentiable at $(0,0)$?


No. If it is differentiable then the derivative is necessarily given by $A(x,y)=x+y$. (By looking at the partial derivatives). To get a contradiction from this consider $\frac {f(x,y)-A(x,y)} {\sqrt {x^{2}+y^{2}}}=\frac {\sin (x^{3}+y^{3}) -(x+y)} {\sqrt {x^{2}+y^{2}}}$. I will let you show that $\frac {\sin (x^{3}+y^{3})} {\sqrt {x^{2}+y^{2}}} \to 0$ and $\frac {x+y} {\sqrt {x^{2}+y^{2}}} $does not tend to $0$.


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