Is this function differentiable?

Question

Is f(x,y)=sqrt(x^2+y^2) differentiable at (0,0)? Justify.

 

I don't think the partial derivatives exist due to the limit definition, so can I say that the function isn't differentiable? Or is that not enough?

Answer 1

Let's apply formal definition
if partial derivative wrt X exists then at (x*,0)
f'(x*,0) = lim{(x,y)->(x*,0)}|f(x,0)-f(x*,0)|\|x-x*|

 

quotient is 

[sqrt(x^2)-sqrt(x*2)]\[x-x*]

=

[|x|-|x*|]\[x-x*].

 

as we approach x*=0 from the right |x| = x so lim= 1

from the left |x| = -x so lim = -1

since limits are different it does not exist and thus is not differentiate at 0,0

 

hope helpful best - David