Editing Floor and ceiling functions (section)

===Continuity and series expansions===
None of the functions discussed in this article are [[continuous function|continuous]], but all are [[piecewise linear function|piecewise linear]]: the functions <math>\lfloor x \rfloor</math>, <math>\lceil x \rceil</math>, and <math>\{ x\}</math> have discontinuities at the integers.

<math>\lfloor x \rfloor</math> is [[semi-continuity|upper semi-continuous]] and <math>\lceil x \rceil</math> and <math>\{ x\}</math> are lower semi-continuous.

Since none of the functions discussed in this article are continuous, none of them have a [[power series]] expansion. Since floor and ceiling are not periodic, they do not have uniformly convergent [[Fourier series]] expansions.  The fractional part function has Fourier series expansion<ref>Titchmarsh, p. 15, Eq. 2.1.7</ref>
<math display="block">
\{x\}= \frac{1}{2} - \frac{1}{\pi} \sum_{k=1}^\infty
\frac{\sin(2 \pi k x)} {k}
</math>
for {{mvar|x}} not an integer.

At points of discontinuity, a Fourier series converges to a value that is the average of its limits on the left and the right, unlike the floor, ceiling and fractional part functions: for ''y'' fixed and ''x'' a multiple of ''y'' the Fourier series given converges to ''y''/2, rather than to ''x''&nbsp;mod&nbsp;''y''&nbsp;=&nbsp;0. At points of continuity the series converges to the true value.

Using the formula <math>\lfloor x\rfloor = x - \{x\}</math> gives
<math display="block">
\lfloor x\rfloor = x - \frac{1}{2} + \frac{1}{\pi}  \sum_{k=1}^\infty \frac{\sin(2 \pi k x)}{k}
</math> 
for {{mvar|x}} not an integer.