Bounded Variation Separates Weak and Strong Average Lipschitz

We closely examine a recently introduced notion of average smoothness. The latter defined a <i>weak</i> and <i>strong</i> average-Lipschitz seminorm for real-valued functions on general metric spaces. Specializing to the standard metric on the real line, we compare these notions to bounded variation (BV) and discover that the weak notion is strictly weaker than BV while the strong notion is strictly stronger. Along the way, we discover that the weak average smooth class is also considerably larger in a certain combinatorial sense, which is made precise by the fat-shattering dimension.