Let (X,ρ,μ) be a metric measure space of homogeneous type, and let Cc⁎(X) denote the set of all Lipschitz functions f on X such that limr→0+supy∈B(⋅,r)|f(⋅)−f(y)|/r converges uniformly to a compactly supported continuous function Lip f on X. Let p∈[1,∞) and let f∈Cc⁎(X) be such that the pair (f,Lipf) satisfies a certain Poincaré inequality. Assume that α∈R∖{0} if p∈(1,∞), and α∈(0,∞) if p=1. Under these assumptions, the authors prove the following result which extends a recent formula of H. Brezis, A. Seeger, J. Van Schaftingen, and P.-L. Yung on finite dimensional Euclidean spaces:supλ∈(0,∞)λp∬Dλ[U(x,y)]α−1dμ(x)dμ(y)∼∫X[Lipf(x)]pdμ(x), whereDλ:={(x,y)∈X×X:|f(x)−f(y)|>λρ(x,y)[U(x,y)]αp} for any λ∈(0,∞), V(x,y):=μ(B(x,ρ(x,y))) and U(x,y):=min{V(x,y),V(y,x)} for any x,y∈X, and the positive constants of equivalence are independent of f. A similar result remains true if p∈(1,∞) and f lies in a wider class of functions in the Hajłasz–Sobolev space. As an application, the authors establish new fractional Sobolev and Gagliardo–Nirenberg inequalities on X. These results are applicable to many classical examples, such as Euclidean spaces, with weighted Lebesgue measure, and complete finite dimensional Riemannian manifolds, with non-negative Ricci curvature.