Is it possible to show following by using a point-and-shoot camera?
shallow depth of field
deep depth of field
I'll offer a (partially) dissenting opinion. Point and Shoots, it's true, don't have the ability to shoot as narrow a DOF as most DSLR cameras, both because of the sensor size and the aperture of their lenses.
That doesn't mean you can't get the effect you're looking for in your photos, though -- it just means you've got to be a little more deliberate in setting up your shot. The following shots were all made with P&S cameras:
All three of these had some things in common -- they were zoomed in, and they were all subjects that had some physical separation from their backgrounds. This allows even a P&S to show some background blurring, even if it's not the super-creamy bokeh of a 50mm f/1.2 or a 70-200 f/2.8 on a full-frame DSLR.
(edit) - For a more detailed look at DOF on a P&S, look at this online DOF calculator, which will let you select your camera from a list and shows actual DOF at various apertures & focal lengths. The basic idea here is that even though P&S cameras don't have the performance envelope that a DSLR generally has, all of these cameras have DOF capabilities that you can exploit if you understand how DOF works. You can use your camera controls and compositional techniques to make the most of whatever envelope you've got once you understand the formula.
No for shallow depth of field under most common circumstances. It is possible for at very short focus distances, at least less than 1m (3').
Yes for deep depth of field.
In case you are curious, this is an indirect result of the small sensors they use. Smaller sensors require short focal-lengths which give more extensive depth-of-field for the same angle-of-view. So even though a point and shoot can have a 28-140mm equivalent angle-of-view, for example, the actual lens can be a 4-20mm lens.
A few advanced compact cameras (they have full manual controls, not just P&S) now use slightly larger sensors and have F/1.8 lenses which helps have a relatively shallow depth of field but does not compare to what you can get on a DSLR. See for example the Olympus XZ-1 and Nikon P300.
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Shallow depth of field is hard to achieve with most point-and-shoots for most shots, since their sensors and lens focal lengths tend to be small. There are some exceptions though, based on a fast lens and film or bigger sensor.
However, if the point-and-shoot has macro capability (i.e. is able to focus very close) or has some way to attach a close-up lens, a shot focused very close (a macro shot) will have quite shallow depth-of-field.
For static or semi-static scenes, you could also use Brenizer method of shooting multiple images at a longer focal length and stitching them into a panorama. Thanks to the longer focal length, this would result in a shallower depth of field than taking the same picture at wider angle.
Since deep depth of field can be achieved simply by focusing to infinity, there are hardly any cameras which cannot show it. Even fixed-focus point-and-shoots are focused so that infinity is reasonably in focus.
The only caveat is that to get shallow depth of field with a small sensor you need to be focussing relatively close (which only works for small subjects) or be at the long end of a superzoom lens.