Contextuality is a feature of quantum correlations. It is crucial from a
foundational perspective as a nonclassical phenomenon, and from an applied
perspective as a resource for quantum advantage. It is commonly defined in
terms of hidden variables, for which it forces a contradiction with the
assumptions of parameter-independence and determinism. The former can be
justified by the empirical property of non-signalling or non-disturbance, and
the latter by the empirical property of measurement sharpness. However, in
realistic experiments neither empirical property holds exactly, which leads to
possible objections to contextuality as a form of nonclassicality, and
potential vulnerabilities for supposed quantum advantages. We introduce
measures to quantify both properties, and introduce quantified relaxations of
the corresponding assumptions. We prove the continuity of a known measure of
contextuality, the contextual fraction, which ensures its robustness to noise.
We then bound the extent to which these relaxations can account for
contextuality, via corrections terms to the contextual fraction (or to any
noncontextuality inequality), culminating in a notion of genuine contextuality,
which is robust to experimental imperfections. We then show that our result is
general enough to apply or relate to a variety of established results and
experimental setups.Comment: 20 pages + 14 pages of appendices, 3 figure