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:R2→R2 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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