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I have often wondered if the focus of a lens at a particular focal length is an exact point or a range within a few millimeters. This becomes all the more important when manually focussing. How far from the camera's point of focus may a subject be and still appear crisp?

There's only one distance that is in sharpest focus. Everything in front of or behind that distance is blurry. The further we move away from the focus distance, the blurrier things get. The questions become: "How blurry is it? Is that within our acceptable limit? How far from the focus distance do things become unacceptably blurry?"

What we call depth of field (DoF) is the range of distances in front of and behind the point of focus that are acceptably blurry so that things still look like they are in focus.

The amount of depth of field depends on two things: total magnification and aperture. Total magnification includes the following factors: focal length, subject/focus distance, enlargement ratio (which is determined by both sensor size and display size), and viewing distance. The visual acuity of the viewer also contributes to what is acceptably sharp enough to appear in focus instead of blurry.

The distribution of the depth of field in front of and behind the focus distance depends on several factors, primarily focal length and focus distance.

The ratio of any given lens changes as the focus distance is changed. Most lenses approach 1:1 at the minimum focus distance. As the focus distance is increased the rear depth of field increases faster than the front depth of field. There is one focus distance at which the ratio will be 1:2, or one-third in front and two-thirds behind the point of focus.

At short focus distances the ratio approaches 1:1. A true macro lens that can project a virtual image on the sensor or film that is the same size as the object for which it is projecting the image achieves a 1:1 ratio. Even lenses that can not achieve macro focus will demonstrate a ratio very near to 1:1 at their minimum focus distance.

At longer focus distances the rear of the depth of field reaches all the way to infinity and thus the ratio between front and rear DoF approaches 1:∞. The shortest focus distance at which the rear DoF reaches infinity is called the hyperfocal distance. The near depth of field will very closely approach one half the focus distance. That is, the nearest edge of the DoF will be halfway between the camera and the focus distance.

We must also remember that hyperfocal distance, like the concept of depth of field upon which it is based, is really just an illusion, albeit a rather persistent one. Only a single distance will be at sharpest focus. What we call depth of field are the areas on either side of the sharpest focus that are blurred so insignificantly that we still see them as sharp. Please note that the hyperfocal distance will vary based upon a change to any of the factors that affect DoF: focal length, aperture, magnification/display size, viewing distance, etc. For why this is the case, please see:

Why did manufacturers stop including DOF scales on lenses?
Is there a 'rule of thumb' that I can use to estimate depth of field while shooting?
How do you determine the acceptable Circle of Confusion for a particular photo?
Find hyperfocal distance for HD (1920x1080) resolution?
Why I am getting different values for depth of field from calculators vs in-camera DoF preview?

As well as this answer to Simple quick DoF estimate method for prime lens

Imagine a wall some distance from your camera ­— a flat wall with no depth, and you're facing it straight on. Lens focus is like that: everything in that exact plane is in focus.

(This is a simplification. For real-world lenses this isn't perfectly flat. In reality, a number of unavoidable optical aberrations keep perfection at bay, but for a basic understanding the idea of a flat plane is good enough.)

So, the sharpest focus is at that plane. Focus points on the camera are used to tell the focus system where to look for increased contrast, and (assuming the autofocus works) everything in the same plane should also be in sharp focus. If you focus closer, it's like bringing the "wall" closer to you; focusing further away is pushing it back. The camera can't focus on multiple points that aren't in the same plane. There's just one focus distance. (Autofocus systems offer multiple focus points, but these are just different possible areas of the frame to focus on. If the objects at those points are at different distances, only one point can ever be in perfect focus.)

But, sharp focus doesn't immediately go from that to unrecognizable blur. Each point which should be sharp on your sensor is actually a tiny circle, and the further away from perfect, the larger that tiny circle becomes. (See What is the "Circle of Confusion?").

If the circle is smaller than you can detect (like, smaller than the pixels on your sensor, or smaller than your eye can see in a final print), something which may be closer or further from the ideal "wall" of sharp focus may be literally indistinguishable from perfect sharpness. Additionally, even outside of the limitations of your camera and imaging system, there's some amount of very slight blur that we're willing to accept as "good enough". This where we get the concept of depth of field. Rather than a plane with no thickness, think of an imaginary thick wall with depth closer and further than the distance you're focused at. Everything in that is "within the depth of field", and therefore considered in focus.

(But don't make the mistake of thinking that this imaginary thick wall has hard borders — it's really a "soft" zone where the borders are a definite judgment call.)

In general, smaller apertures (higher f-numbers) provide more depth of field. (See Technically, why is the out of focus area blurred more when using a bigger aperture?)

Geometric optics teach us to expresses that a lens is only able to form a sharp image of an object at a given focus point. Objects nearer or further as to distance will image as unsharp. However, as a matter of common observation, objects before and behind the distance focused on likely appear sharply focused. The reason is, there exists a span before and behind the point of focus that reproduces acceptably sharp. This span is called “depth-of-field".

The lens handles each point of the subject individually. By point, we are talking about a spot so tiny it has no dimension. The job of the lens is to project an image of this point on the image plane. Because of residual uncorrected aberrations and inadequate focusing, the image of points is never reproduced as points. Instead, all points reproduce as circles. These are called circles of confusion because they have scalloped boundaries and are juxtaposed alongside adjacent circles of confusion.

It is the size of these circles that is the determining factor as to whether we pronounce an image as tack sharp or blurred. If the circles are tiny and below our ability to resolve them as circles, we pronounce the image “sharp”. If these image points are seen as circles and not dimensionless points, we pronounce the image unsharp. For the average person with good vision, viewing a photograph in good light, these circles of confusion must be no larger than 2/100 inches (½ mm). when viewed from 20 inches

This is the stuff of “depth-of-field”.

For each particular wavelength and each front lens radius (assuming perfect radial symmetry of the lenses), focus is an exact distance, given that the sensor is a perfect spherical surface with the center being in the focal plane. Wait, it isn't. And we are not working with monochromatic light either. And if the focus distances for various front lens radii don't match perfectly, we get different performances at different apertures.

Depth of field is calculated assuming a perfect lens model and an acceptable "circle of confusion". The truth is that lenses are not perfect. So the "circle of confusion" will be dwarfed by chromatic longitudinal (and lateral) aberration and spherical aberration and lens convergence problems in practice, all of those determining a range where you could not distinguish meaningful levels of sharpness irrespective of sensor resolution.

The "circle of confusion" based depth of field calculation has the advantage of working from well-determinable numbers (at least when using digital sensors with fixed pixel pitch) but it omits taking into account the quality of the optics. If the optics were able to let light through the entire aperture at all visible wavelengths from a single point at focusing distance converge to a single point on the sensor (which actually has non-zero depth of its photon-converting surface, throwing another spanner in the works), you could talk about exact focusing distances.