Can your NEX do this?
If the final print is small enough, or your desired print resolution is low enough, using fNumbers greater than f/11 with this sensor could provide more DoF without causing diffraction to inhibit your desired print resolution. It would be a shame to always shoot at f/11 or smaller fNumbers, for fear of diffraction that's only a problem at higher enlargement factors and/or higher desired print resolutions.
There is no single fNumber at which diffraction will begin to inhibit a desired print resolution  it varies with your anticipated enlargement factor, because the Airy disks recorded at the sensor (or film) plane with a given fNumber will suffer varying degrees of magnification for different size prints examined at an anticipated viewing distance.
If you intend to make 4x6inch prints or even smaller wallet photos, to be viewed at a distance of 20inches, you'll have a lot less concern for the possibility of either diffraction or defocus inhibiting whatever you personally consider to be an acceptable level of subject detail, than you would when making larger prints. You can feel free to shoot at just about any aperture available on your lens when the enlargement factor is small, the viewing distance is great, or your personal requirement for resolution in the final print is low.
If instead you intend to make 8x12inch prints, that must survive scrutiny at a viewing distance of only 10 inches, you'll likely have a preference for greater resolution in the final print, and will therefore suffer a greater possibility of diffraction or defocus inhibiting the print resolution you desire.
Thus, it's impossible for anyone to recommend an aperture at which "diffraction begins to destroy sharpness" for any given camera, without knowing the enlargement factor at which you plan to print your images and your desired print resolution (or, at the very least, your anticipated viewing distance, so that we can recommend a desired print resolution.)
Again: There is no single fNumber at which diffraction "becomes a problem" for any camera at all combinations of enlargement factor and desired print resolution.
Note that an image for an enormous roadside billboard does not have to be shot with a 10,000 Megapixel camera because the associated viewing distance is typically hundreds of feet. A 24x36inch print viewed at a distance of 20 inches can similarly appear to have every bit as much subject detail as when the same file is printed to a 12x18inch print for viewing at half that distance.
Yet somehow, anticipated enlargement factor and specification of the resolution one personally hopes to record in the final print are, more often than not, completely ignored in discussions of aperture selection for controlling either diffraction or DoF.
A good number of DoF calculators don't even allow userspecification of a custom CoC diameter, instead allowing the user to specify only the focal length plus the near and far distances of the subject space. Countless people have been disappointed by the results had when they make use of the DoF scales engraved on their lens barrels. Why are they disappointed? Because enlargement factor and the amount of resolution they personally hoped to secure in the final image (which itself should take viewing distance into consideration) are ignored by such tools!
Have a look at this formula from Wikipedia's Circle of Confusion page, http://en.wikipedia.org/wiki/Circle_of_confusion
CoC (mm) = viewing distance (cm) / desired finalimage resolution (lp/mm) for a 25 cm viewing distance / enlargement / 25
For example, to support a finalimage resolution equivalent to 5 lp/mm for a 25 cm viewing distance when the anticipated viewing distance is 50 cm and the anticipated enlargement is 8:
CoC = 50 / 5 / 8 / 25 = 0.05 mm
Notice that the only variables in this equation for calculating the maximum CoC diameter to be used in DoF calculations are those for viewing distance, desired final image resolution (in lp/mm), and enlargement factor, but again, these variables are seldom proclaimed when discussing aperture selection.
A viewing distance of 25 cm is about 9.84 inches  about the closest that a person with healthy vision can focus with the naked eyes. If you are willing to confine your specification of a desired final image resolution to always satisfy a viewing distance of 25cm, you can reduce the Max. CoC calculation to this equation:
CoC (mm) = 1 / desired finalimage resolution (lp/mm) for a 25 cm viewing distance / enlargement factor
Here's a similar formula for determining the aperture, f/N, at which diffraction's Airy disks will begin to inhibit your desired final image resolution (in lp/mm) at the anticipated enlargement factor for a 25cm viewing distance:
N = 1 / desired print resolution (lp/mm) / anticipated enlargement factor / 0.00135383
Notice that it's just the calculated CoC value divided by the constant 0.00135383.
See David M. Jacobson's Lens Tutorial for the origin of this constant: http://graflex.org/lenses/photographiclensestutorial.html .
This same math is employed by Sean McHugh's diffraction tutorial at http://www.cambridgeincolour.com/tutorials/diffractionphotography.htm .
(Continued below...)
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