real analysis – A non-differentiable function $f(x,y)$ with bounded $f_x$, $f_y$, $f_{xx}$ and $f_{yy}$

Recently I was trying to construct a counterexample to the statement “If there exist $f_{xy}(0,0)$, $f_{yx}(0,0)$ and the functions $f_{xx}$, $f_{yy}$ exist in some neighborhood and are continuous at $(0,0)$, then $f$ is twice differentiable at $(0,0)$“. In order to do that the following question arose:

Is there a function $fcolon mathbb{R}^2tomathbb{R}$ with bounded $f_x$, $f_y$, $f_{xx}$, $f_{yy}$, which is non-differentiable at $(0,0)$?

If there exists such a function, then we’re done. The functions which came to mind have unbounded second partials $f_{xx}$ and $f_{yy}$.

urr = fxx cos ^ 2x + 2fxy cosx sinx + fyy sin ^ 2 x

Enter the image description here

For this question applies, if Ur should take place after the first partial derivation. It will result in a function Ur (x, y, θ). Why do I have to consider the following Urr values ​​the second time I make a partial derivative: Urr = ∂Ur / ∂x * ∂x / r + ∂ Ur / ∂y * ∂y / ∂r + ∂Ur / ∂θ * ∂θ / ∂r? Instead, the answer was Urr = ∂Ur / ∂x * ∂x / ∂r + ∂Ur / ∂y * ∂y / ∂r. Do not we have to take the last term into account? and if so, how would I set up a function to perform the partial derivative of ∂θ / ∂r?
Many Thanks