Disjoint continuous functions

We define analogues of supports of continuous functions to general Hausdorff spaces and disjointness relations for such functions, and prove that this data completely determines locally compact spaces satisfying certain regularity conditions. This allows us to recover general classical results of similar nature by Kaplansky, Banach-Stone, Milgram, and a version of a result by Wendel, more recent results by Jarosz, Li-Wong, Hernandez-Rodenas, a classification of diagonal-preserving automorphisms of Steinberg algebras (with trivial cocycles) and of Renault's convolution groupoid algebras, as well as a new classification of certain isomorphisms of groups of circle-valued functions. To this end, we employ recent results by Bice-Starling (arXiv:1609.09708 [math.GN]}.