Friday, 26 August 2016

analysis - Can I say f is differentiable at c if Du(c)=nablaf(c)cdotu for all unit vectors u?



Can I say f is differentiable at c if Du(c)=f(c)u for all unit vectors u?



I think I can because it guarantees a tangent plane.



But I don't know how to prove this precisely.




Anyone can give an advice or a proof?



Thanks.


Answer



Now you cannot say this. Consider f:R2R2 given by
f(x)={1x21=x2,x2>00otherwise


at c=0. Then for any ray [0,)u, f is constant on some intervall [0,ϵ)u, that is Duf(0)=0. But f is not continuous at 0, hence not differentiable.



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