Constrained optimisation of function with nonzero gradient

Question

Suppose a continuous function $f(x,y,z):\mathbb{R}\longrightarrow\mathbb{R}$ is such that $\nabla f(x,y,z)\neq\mathbf{0}$ for every $(x,y,z)\in [0,1]^3$. A property of the gradient is that it is zero at stationary points. Thus one can say that there is no stationary point in the interior of the cube. The cube is compact, so $f$ must have minimum and maximum on it. Hence these must be on the boundary of the cube. Suppose the restrictions of $f$ on each face of the cube share the same property, that is $\nabla_{x,y}f(x,y,0)\neq \mathbf{0}$ etc. I believe the reasoning then iterates and one can conclude that the maximum and minimum must occur on the edges of the cube. For a function having such property, constrained maximisation and minimisation is thus reduced to the standard univariate study of the maxima and minima of the restrictions of $f$ on the edges of the cube. Is there anything wrong with this reasoning? Thanks for any opinion you can share.