How would the condition for linearity for function with multiple variable be given? I know that a function is called linear if it satisfies the conditions
$$f(x+y)=f(x)+f(y)$$
and
$$f(ax)=af(x)$$
(i.e. it preserves the properties or operation).
How is the condition for linearity given when there are multiple variable in the funciton?
for ex
$$f(x+y,z)$$ and 
$$f(ax,z)$$
I know
$$f(x+y,z)=f(x,z)+f(y,z)$$ 
is a condition for multilinear function, but can there be multivariable function which is linear and not multilinear,ie linear in all the variables at once and not seperately (like in multilinear function). How to make sense of multivariable function that is linear and that is multilinear? 
Edit: Extra details.
A function $f(x,y)$ is called multlinear(bilinear to be specific) if
$$f(x+a,y)=f(x,y)+f(a,y)$$
$$f(x,y+b)=f(x,y)+f(x,b)$$
and also $$f(nx,y)=nf(x,y)=f(x,ny)$$.
So is 
$$f(x+a,y+b)=f(x,y+b)+f(a,y+b)+f(x+a,y)+f(x+a,b)+f(x,y)+f(a,b)+f(x,b)+f(a,y)$$ 
when $f(x,y)$ is multilinear and
$$f(x+a,y+b)=f(x,y)+f(a,b)$$ 
when $f(x,y)$ is linear?
 A: Let $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ a function. We say $f$ is linear in the $i^{th}$ variable if, given fixed $x_{j}$ for $j\neq i$ the function $T$ defined by 
$$T(x)=f(x_{1},x_{2},\dots,x_{i-1},x,x_{i+1},\dots,x_{n}) $$
is linear. If $f$ is linear in the $i^{th}$ variable for each $i$, we say that $f$ is multilinear.
For example, an inner product <$\cdot,\cdot$> is bilinear, because
$$\left<x+y,z\right>=\left<x,z\right> + \left<y,z\right> $$
$$\left<x,y+z\right>=\left<x,y\right> + \left<x,z\right> $$
A: If the function $\, f \,$ is the sum of linear functions in each variable separately then you can call the function linear. For example, if
 $\, f(x,y,z) := ax + by + cz, \,$ then $\, f(nx,ny,nz) = n \, f(x,y,z) \,$ and
 $$\, f(x_1\!+\!x_2, y_1\!+\!y_2, z_1\!+\!z_2) = a(x_1\!+\!x_2) \!+\! b(y_1\!+\!y_2) \!+\! c(z_1\!+\!z_2) = f(x_1,y_1,z_1) \!+\! f(x_2,y_2,z_2). \,$$
An example of a bilinear function is $\, f(x,y) = axy \,$ where $\, f(nx,y) = f(x,ny) = n\, f(x,y) \,$ and
 $$\, f(x_1+x_2,y_1+y_2) = f(x_1,y_1) + f(x_1,y_2) + f(x_2,y_1) + f(x_2,y_2). \,$$
