Depth of field is a perception phenomenon which factors in the HVS (human visual system). It is really a game of "how much blur can we have until it becomes objectionable?" There is only one "plane" (usually really a segment of a sphere) which is in focus. At that point the imaging system performs in accordance with losses such as atmospherics and the MTF (modulation transfer function) of the lens.
As an object moves off that plane, it immediately becomes "out of focus" and there is a point spread function which describes a growing disk which is in some circles (no pun intended) called the "circle of confusion."
Smaller apertures employing central portions of the lens, have the light taking a shorter (and more consistent) path through the lens. This helps reduce the point spread function which describes the circle of confusion (and not always a circle). The point spread function of an optics system is also called the impulse response.
The resultant image is one which is the convolution of the target image and the point spread function. At least for non-coherent imaging. So the perception of the depth of field is linear with the f-stop and focal length.
Unfortunately, depth of field has it limits, and a very very small aperture will not provide nearly infinite depth of field, because diffraction plays a greater role, in blurring the image, as the aperture gets smaller.
So what really happens with depth of field is that objects are not really in focus off the focused plane, but rather the blur is considered negligible. Think of it this way: a thumbnail photo might look clear, but if expanded to be an 8x10" photo, it may be unacceptably fuzzy. So acceptable depth of field is a determination of the effect of the impact of an off focused image on the observer, given the optical system (atmospherics, lens, sensor/film, and rendering/printing process) and the perception perspective (how big is the viewed image).
In practical application, a so-called hyper-focal setting on a lens, may give an acceptable image of a scene when viewed on a small format display or print, but when expended or enlarged, will yield a more fuzzy appearance as it is in reality not completely in focus through the "depth of field."