  Atomically-resolved imaging and force measurements using the atomic force
microscope (AFM) are performed most commonly in a frequency-modulation (FM)
mode. This has led to spectacular results, including direct observation of the
atomic structure of complex molecules and quantification of chemical and
frictional forces at the atomic scale. We address here a critical question: Is
recovery of force from the measured frequency shift experienced by the AFM
cantilever ill-posed - that is, unreliable in the presence of (unavoidable)
measurement uncertainty? Resolution of this issue underlies all force
measurements using FM-AFM, but remains outstanding. It is shown that concavity
of the force law's distance dependence controls ill-posed behavior, with a
rapid concavity change corrupting force measurements by inducing spurious and
unphysical effects - such rapid change is not uncommon. Practical conditions to
eliminate ill-posed behavior are formulated which are verified experimentally.
This study lays the foundations for robust atomically-resolved force
spectroscopy and future work that will seek to regularize ill-posed force
measurements.
