Fractional part
From Calculus
This article is about a particular function from a subset of the real numbers to the real numbers. Information about the function, including its domain, range, and key data relating to graphing, differentiation, and integration, is presented in the article.
View a complete list of particular functions on this wiki
Definition
The fractional part is a function from the reals to the reals that sends every real number to where denotes the greatest integer function of .
Key data
Item | Value |
---|---|
default domain | all real numbers |
range | , i.e., the set absolute maximum value: does not exist, but the lub of the range is 1, absolute minimum value: 0 |
continuity | The function is continuous everywhere except at integers. At any integer, the left hand limit of the function is 1, and the right hand limit is 0. In particular, the function is right continuous at all points (including integers) and not left continuous at integers. |
period | The function has a period of 1. |
derivative | the function is differentiable everywhere except at integers, and at all points other than integers, the derivative is 1. at integers, the right hand derivative is 1, and the left hand derivative does not exist. |
antiderivative | Fill this in later -- a piecewise parabola |