calculus - Prove function is not differentiable even though all directional derivatives exist and it is continuous.

In the following link, the function below is provided as an example of a function being continuous and all directional derivatives existing. Yet, it is not differentiable. How do I prove that this is not differentiable?

$$
f(x,y)=
\begin{cases}
\frac{y^3}{x^2+y^2},& \text{if } (x,y) \neq (0,0)\\
0, & \text{otherwise}
\end{cases}

$$

Answer

The definition of the directional derivative at $0$ is
$$df(v)=\lim\limits_{t\to 0}\frac{f(tv)-f(0)}{t}$$ if $v=(a,b)$ then

$$\frac{f(tv)-f(0)}{t}=\frac{\frac{t^3a^3}{t^2a^2+t^2b^2}}{t}=\frac{a^3}{a^2+b^2}$$

So the directional derivative is
$$df(a,b)=\frac{a^3}{a^2+b^2}$$
Now the condition for differentiablity is that $df$ is a linear function. This is not the case so the function is not differentiable, although all directional derivatives exist.

I had a minor realization about this recently, if you take for example $z=f(x,y)$ and graph it as a surface and if directional derivatives of $f$ exist then then the surface will have slopes in all directions. So there are well defined tangent lines in all directions. However the function is differentiable only if all those tangent lines lie on the same plane. If you graph this function in wolfram alpha you can see that this is not the case, as was also shown above.