Geometric meaning of equal partial derivatives Suppose that $f\in C^1(\mathbf{R}^3)$ and $$\frac{\partial f}{\partial x}=\frac{\partial f}{\partial y}=\frac{\partial f}{\partial z}$$
By mean value theorem and the condition above, we have
$$ f(x,y,z)-f(x+y+z,0,0)=0$$ 
So values of $f$ on the whole space have the corresponding value on the $x$-axis, but the mean value theorem approach above is rather not intuitive.
Could we dig some geometric(or intuitive) meaning out of the condition above?
 A: It seems that I do not have enough reputation for commenting. I understand it the way that with geometric meaning you mean how such functions look like. One first step would be to look at some examples(which is a bit complicated for functions $\mathbb{R}^3 \to \mathbb{R}$). Assume first one space dimension less, then you can look at $f(x,y) = constant$ and $f(x,y) = e^{(x + y)}$ for some simple examples. The latter example is one which I would use to get a first geometric understanding for your case of a function $\mathbb{R}^3 \to \mathbb{R}$:
The gradient is known to give you the directions of steepest descent and steepest ascent. Note that for the functions $f$ you are interested in $\nabla f(a,b,c) \in S := \operatorname{span} \left\{ (1,1,1)^t \right\}$ for every point $(a,b,c)^t \in \mathbb{R}^3$. So in the simplified example $f(x,y) = e^{(x + y)}$ in the directions orthogonal to $S_2 := \operatorname{span} \left\{ (1,1)^t \right\}$ the function $f$ remains constant, parallel to $S_2$ in the "positive" direction there is a rapid increase resulting in something looking like a wall of inifinite length rising with exponentially increasing hight to the sky and in the "negative" direction everything flattens out to 0$.
A bit more interesting example to look at is $f(x,y) = sin(x+y)$. Whats happening is of course the same than before, just now you get periodic behaviour propagating in parallel direction of $S_2$. 
I think a good summary would be that geometrically such functions f are always  copies of the graph of a function $\mathbb{R}^2 \to \mathbb{R}$ stapled parallel(it is hard to imagine 4D!) to the orthogonal complement of $S$. (The formal proof would maybe like in the less dimensional case use, as indicated above, that the gradient gives you the "level sets" and the directions of steepest descend)
In the more imaginable way the simple example $f(x,y) = sin(x+y)$ are infinite copies of  the graph of a sinus function $\mathbb{R} \to \mathbb{R}$ clued next to each other, like the amunition for your office stapler, where each needle represents one of these graphs.
