Showing that the function given by $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$ and $f(0,0)=0$ is continuous but not differentiable 
Let
$$
 f(x,y) =
\begin{cases}
\dfrac{xy}{\sqrt{x^2+y^2}}   & \text{if $(x,y)\neq(0,0)$ } \\[2ex]
0 & \text{if $(x,y)=(0,0)$ }  \\
\end{cases}
$$
Show that this function is continuous but not differentiable at $(0,0),$ although it has both partial derivatives existing there.


I can show this function is continous and the partial derivatives exist. But how can I show that this function is not differentiable?
Is showing that the function is differentiable similar to showing that a derivative exists?
 A: There are no directional derivatives in nearly all directions. Consider, in particular, along the line $y=x$.  $f(x,y)$ is a constant times the absolute value function.

When a function of two variables is differentiable, then there is a tangent plane to the surface $z=f(x,y)$, and there are directional derivatives in all directions.  This one doesn't have directional derivatives except in two directions, and there's no tangent plane to the surface $z=f(x,y)$.
A: Since $|f|\le|x|$, $\lim_{(x,\,y)\to(0,\,0)}f=0$. But $\frac{\partial f}{\partial x}=\frac{y^3}{(x^2+y^2)^{3/2}}$, which $\to1$ along an $x=0,\,y\to0$ path, or $\to0$ along an $x\to0,\,y=0$ path.
A: I don't know if I should write my answer here but can't write all this as a comment.
 This is what I tried to do:
According to the definition , $f$ is fifferentiable at $a $ if,
$\Delta$f=$f$(a$_1$+h$_1$,...,a$_n$+h$_n$)-$f$(a$_1$,a$_2$,..,a$_n$)
=l$_1$h$_1$+l$_2$h$_2$+...+l$_n$h$_n$+...+B$_1$h$_1$+...+B$_n$h$_n$
where l's are constants and $ \lim_{h1h2...hn\to 0,0,,..,0}$B$_i$=0  
Here a=(0,0,).
Therefore  $\Delta$f=f(0+h$_1$,0+h$_2$)-f(0,0)
=$\frac{h_1h_2}{\sqrt{h_1^2+h_2^2}}$-0
This is not in the above form .Therefore f is not differentiable at (0,0)
A: CHECK IT, it's easier one! U must determine the partial derivatives first.
function of two Variable $f(x, y)$ is disterentiable at point $(a,b)$ it $(h, k) \rightarrow(0,0) \quad \frac{f(a+h, b+k)-f(a, b)-h f_{x}(a, b)-k f_{y}(a, b)}{\sqrt{h^{2}+k^{2}}}$
$f_{x} \rightarrow$ partial derivative wrt $x$ i.e
$f(x, y)=\frac{x y}{\sqrt{x^{2}+y^{2}}} ;
 \quad \quad f_{x}=\operatorname{lt} \frac{f(x, 0)-f(0,0)}{x}$
$\begin{aligned}(x, b) \rightarrow(0,0) & \text { at }(0,0) \\ \text { accordingly } & f_{y}=0 \end{aligned}$ lt$(h, k) \rightarrow(0,0) \quad \frac{f(h, k)-f(0,0)-h f_{x}(0,0)-k f_{y}(0,0)}{\sqrt{h^{2}+k^{2}}}$
$=\frac{\frac{h K}{\sqrt{h^{2}+K^{2}}} \ -0-0-0}{\sqrt{h^{2}+K^{2}}}=\frac{h K}{\left(h^{2}+K^{2}\right)}$
along the path $k=m h$
$\lim _{h \rightarrow 0} \frac{m h \cdot h}{h^{2}+m^{2} h^{2}}=\log _{k \rightarrow 0} \frac{m h^{2}}{h^{2}\left(1+m^{2}\right)}=\frac{m}{1+m^{2}}$
Limiting value is not unique.
Hence not differentiable though it is continous
