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mahler1

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Homework Statement .

Let ##f:\mathbb ℝ:→ℝ^2## be a function defined as:

##f(x,y)=\frac{x^2y-2xy+y} {(x-1)^2+y^2} \forall (x,y)≠(1,0)## and ##f(1,0)=0##.

Prove that for any curve ##α:(-ε,ε)→ℝ^2## of class ##C^1## (where ##ε>0##) such that ##α (0)=(1,0)## and ##α(t)≠(1,0)## for every ##t≠0##, the derivative ##(foα)'(0)## exists; but that ##f## is not differentiable at the point ##(1,0)##.

The attempt at a solution:

I didn't have any problems to prove that f isn't differenitable at ##(1,0)##. First, I proved that the partial derivatives exist. Both of them gave 0, then supposed ##f## was differentiable at that point. If this was the case, the limit ##\lim_{(x,y)\rightarrow (1,0)} \frac{ |f(x,y)-(f(1,0)+<∇f(1,0),(x-1,y)>|} {||(x-1,y)||}## would have to equal 0. If I consider the curve ##ψ(t)=(t+1,t)##, then the limit is ##\frac{\sqrt(2)} {4}≠0##.

I got stuck in proving the first part of the statement "for any curve ##α:(-ε,ε)→ℝ^2##..., the derivative ##(foα)'(0)## exists". I mean, I can't show this using the limit definition: ##\lim_{h\rightarrow 0} \frac {foα(h)-foα(0)} {h}##, because I don't have enough information to calculate it. Am I doing something wrong? What should I do to prove that part of the statement?

Let ##f:\mathbb ℝ:→ℝ^2## be a function defined as:

##f(x,y)=\frac{x^2y-2xy+y} {(x-1)^2+y^2} \forall (x,y)≠(1,0)## and ##f(1,0)=0##.

Prove that for any curve ##α:(-ε,ε)→ℝ^2## of class ##C^1## (where ##ε>0##) such that ##α (0)=(1,0)## and ##α(t)≠(1,0)## for every ##t≠0##, the derivative ##(foα)'(0)## exists; but that ##f## is not differentiable at the point ##(1,0)##.

The attempt at a solution:

I didn't have any problems to prove that f isn't differenitable at ##(1,0)##. First, I proved that the partial derivatives exist. Both of them gave 0, then supposed ##f## was differentiable at that point. If this was the case, the limit ##\lim_{(x,y)\rightarrow (1,0)} \frac{ |f(x,y)-(f(1,0)+<∇f(1,0),(x-1,y)>|} {||(x-1,y)||}## would have to equal 0. If I consider the curve ##ψ(t)=(t+1,t)##, then the limit is ##\frac{\sqrt(2)} {4}≠0##.

I got stuck in proving the first part of the statement "for any curve ##α:(-ε,ε)→ℝ^2##..., the derivative ##(foα)'(0)## exists". I mean, I can't show this using the limit definition: ##\lim_{h\rightarrow 0} \frac {foα(h)-foα(0)} {h}##, because I don't have enough information to calculate it. Am I doing something wrong? What should I do to prove that part of the statement?

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