How to see that this map is not smooth? Let $f(x,y) = x^2$ and define $g(x,y) = {x \cdot f(x,y) \over x^2 + y^2 } = {x^3 \over x^2 + y^2}$.
In this book here it is stated (on page 86) that this function does not extend to a smooth function in a neighbourhood of $(0,0)$. 
But I don't see why $g$ should not be smooth on all of $x$. I took a derivative but that didn't yield any insight and then I plotted it for different min values of x,y (here) and it looks smooth:


Please could someone explain to me why $g$ is not smooth on $\mathbb
 R^2$?


It is not smooth as it is at least not differentiable at $(0,0)$. Consider the directional derivatives at the origin in the directions $(1,\lambda)$.
