According to this article:
The gradient of a function w=f(x,y,z) is the vector function:
$<\dfrac{\partial f}{\partial x} (x, y, z), \dfrac{\partial f}{\partial x} (x, y, y),\dfrac{\partial f}{\partial z} (x, y, z)>$
Isn't this a vector? Why do we call it a function?
Because it depends on $(x,y,z)$.
(Such a function, which assigns a vector to each point $(x,y,z)$, is also called a vector field.)