Let $f: \mathbb{R}^2 \to \mathbb{R}$ $$f = \begin{cases}
\frac{x^3+y^3}{\sqrt{x^2+y^2}} & (x,y) \ne 0 \\
0 & (x,y)=0 \\
\end{cases}$$ show that $f$ is differentiable at the origin.
What I did was just use the definition of a partial derivative.
For $\frac{\partial}{\partial x}(0,0) = \lim_{h\to 0} \frac{f(x+h,y)-f(x,y)}{h} = \lim_{h\to 0} \frac{f(h,0)-f(0,0)}{h} = \lim_{h\to 0} \frac{h^3}{\sqrt{h^2}} = h^2 = 0$
and similarly for $\frac{\partial}{\partial y}(0,0)$. Is this enough to show differentiability or do I have to show something else also? For $f$ to be differentiable at the origin it would satisfy if it has continuous partial derivatives at the origin? The partials certainly exist, but I'm not sure about continuity here. Also I guess this could be shown by polar coordinates, however I'm not familiar with them so I would like to use other methods.
Best Answer
Before showing differentiability you need to show continuity in $(0,0)$.
We are in $\mathbb R^2$, so we have to select a norm, and given the denominator of $f(x,y)$ it seems appropriate to choose the Euclidean norm$$||(x,y)||_2=\sqrt{x^2+y^2}$$
So notice $|x^3|<x^2$ and $|y^3|<y^2$ when $x,y$ are small.
$$|f(x,y)|\le\dfrac{|x^3|+|y^3|}{\sqrt{x^2+y^2}}\le \dfrac{x^2+y^2}{\sqrt{x^2+y^2}}=\sqrt{x^2+y^2}=||(x,y)||_2\to 0$$
Therefore $f$ is continuous in $(0,0)$ and $f(0,0)=0$.
Now for differentiability you need to evaluate
$\dfrac {|f(x,0)-f(0,0)|}{|x|}=\dfrac{\frac {x^3}{|x|}-0}{|x|}=\dfrac{x^3}{x^2}=|x|\to 0$
By symmetry we have the same for $y$ and set $f_x(0,0)=0$ and $f_y(0,0)=0$.
Now we look at the differentiability:
$\begin{align}\dfrac {|f(0+x,0+y)-\overbrace{f(0,0)}^0-\overbrace{f_x(0,0)}^0x-\overbrace{f_y(0,0)}^0y|}{||(x,y)||_2}=\dfrac{|x^3+y^3|}{x^2+y^2}\\\\\le\dfrac{x^2|x|+y^2|y|}{x^2+y^2}\le \dfrac{x^2+y^2}{x^2+y^2}\max(|x|,|y|)\end{align}$
The last part is obtained via $\max(|x|,|y|)=||(x,y)||_\infty\le||(x,y)||_2\to 0$