# Testing Piecewise Functions

## TL;DR

This paper investigates the query complexity of property testing for general piecewise functions on the real line, revealing that active testing complexity is independent of the number of pieces, with optimal bounds identified for passive testing of piecewise constant functions.

## Contribution

It introduces a zero-measure crossings condition and analyzes the query complexity for both active and passive testing of piecewise functions, highlighting independence from the number of pieces in active testing.

## Key findings

- Active testing complexity is independent of the number of pieces.
- Passive testing complexity depends on the number of pieces for piecewise constant functions.
- Optimal bounds are established for passive testing in specific cases.

## Abstract

This work explores the query complexity of property testing for general piecewise functions on the real line, in the active and passive property testing settings. The results are proven under an abstract zero-measure crossings condition, which has as special cases piecewise constant functions and piecewise polynomial functions. We find that, in the active testing setting, the query complexity of testing general piecewise functions is independent of the number of pieces. We also identify the optimal dependence on the number of pieces in the query complexity of passive testing in the special case of piecewise constant functions.