Lineability theory asks whether a set of pathological functions contains a large linear or algebraic structure. For every s in (1, 2] the continuous functions on [0, 1] whose graph has Hausdorff and box dimension both equal to s form a strongly c-algebrable set, obtained by composing the Weierstrass function with exponential-like generators inside the Hölder space of exponent 2 - s and invoking bi-Lipschitz invariance of Hausdorff dimension. The parallel question for pointwise regularity behaves differently: the functions in a Hölder space whose pointwise Hölder exponent is constant and maximal at every point are c-lineable but not 1-algebrable, since powers of an algebra generator become arbitrarily smooth at any prescribed point. Relaxing 'every point' to 'outside a set of Hausdorff dimension zero' restores strong c-algebrability, via a strongly monoHölder function all of whose level sets have dimension zero, and the same construction yields strongly c-algebrable families of functions that are nowhere Riemann-Liouville fractional differentiable above the Hölder exponent while fractional differentiable below it.
This paper by Esser, Maghsoudi, Rodriguez-Vidanes and Seoane-Sepulveda applies lineability theory to families of continuous real functions on the unit interval that exhibit extreme irregularity, aski…