Nowhere Differentiability of the Generalized Takagi Function at the Critical Threshold

This paper provides a complete resolution of the regularity trichotomy for the generalized Takagi function Tα,β(x)=∑n=0∞αnτ(βnx)Tα,β(x)=∑n=0∞αnτ(βnx), where τ(x)=dist⁡(x,Z)τ(x)=dist(x,Z).

We prove that at the critical threshold αβ=1αβ=1, the function exhibits a distinctive logarithmic failure of differentiability. Specifically, for every point xx, the difference quotients grow asymptotically like Θ(∣log⁡h∣)Θ(∣logh∣) as h→0h→0, confirming that the function is continuous but nowhere differentiable in this regime.

This result finalizes the classification:

• Subcritical (αβ<1αβ<1): Lipschitz continuous.

• Critical (αβ=1αβ=1): Continuous, nowhere differentiable with Θ(∣log⁡h∣)Θ(∣logh∣) growth.

• Supercritical (αβ>1αβ>1): Continuous, nowhere differentiable with power-law (Ω(∣h∣−γ)Ω(∣h∣−γ)) growth.

The proof is elementary and leverages the function's self-similarity through a novel telescoping argument and a combinatorial analysis of the base-$\beta$ expansion of $x$. This work establishes the generalized Takagi function as a fundamental example of a phase transition in functional regularity.