Monday, February 5, 2018

multivariable calculus - How do I prove continuity of partial derivative of f(x,y)=sqrtx4+y4 at (0,0)?



Consider I have f(x,y)=x4+y4
And I want to check if the function has partial derivatives continuous in point (x0,y0)=(0,0)



I know theorem, that existence of continuous partial derivatives implies differentiability of this function, and differentiablity implies that function is continuous. I can also test this function for if it is differentiable. Can I use those information to check if partial derivatives are continuous? If not, what is standard method of calculating continuity of partial derivative?



fx=lim



Answer



You have:
f_x(x,y)= \begin{cases} \frac{2x^3}{\sqrt{x^4+y^4}}&(x,y)\neq(0,0)\\ 0&(x,y)=(0,0)\\ \end{cases}
and this function is continuos in \mathbb R^2 because |f_x(x,y)|\leq 2|x| for each (x,y)\in\mathbb R^2.
By simmetry, also f_y is continuous in \mathbb R^2.



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