You can also imagine this geometrically:\\
$z = f(x, y)$ is 3-dimensional. If you fix the variable $x$ you'll make things 2-dimensional (the intersecting plane). If you then fix $y$ you'll get an actual point $z$.
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\includegraphics*[scale=0.4]{./images/Grafico_3d_x2+xy+y2.png}
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For every of these steps we can define a real new function. This scales up to any number of dimensions/arguments.