# On the rate of convergence for Takagi class functions

## TL;DR

This paper studies the convergence rates of generalized Takagi functions using probabilistic laws, revealing that the classic Takagi function does not follow the law of large numbers in the standard way.

## Contribution

It introduces probabilistic conditions for convergence rates of Takagi class functions and demonstrates the non-compliance of the original Takagi function with the law of large numbers.

## Key findings

- Probabilistic laws describe convergence rates of Takagi functions.
- The Takagi function does not satisfy the law of large numbers normally.
- Conditions for convergence include LLN, CLT, and LIL.

## Abstract

We consider a generalized version of the Takagi function, which is one of the most famous example of nowhere differentiable continuous functions. We investigate a set of conditions to describe the rate of convergence of Takagi class functions from the probabilistic point of view: The law of large numbers, the central limit theorem, and the law of iterated logarithm. On the other hand, we show that the Takagi function itself does not satisfy the law of large numbers in the usual sense.