Кусково постійні функції

What is a step function?

A step function is a piecewise function over the real numbers that is made up from a finite number of constant functions. The name comes from the resemblance of the function to a set of steps.

Examples of step functions

$$f(x)=\begin{cases} 0 & x<0 \\ 1 & 0\leq x<5 \\ 2 & x\geq 5 \end{cases}$$

$$g(x)=\begin{cases} 0 & x<-1 \\ 3 & -1\leq x\leq 1 \\ 0 & x>1 \end{cases}$$

$$h(x)=\begin{cases} 3 & x<1\\ 1 & x=1 \\ -1 & x>1 \end{cases}$$

What makes each of the above functions step functions?

• They are discontinuous.
• Each piece of the functions is a constant function.
• They are all functions.

$$\text{Examples of step functions}$$

Floor and ceiling functions

The floor function, $f(x)=\lfloor x \rfloor$ (also written as $\text{floor}(x)$ or $[x]$), takes a real number $x$ and returns the greatest integer less than or equal to $x$. $$\begin{align} \lfloor 2 \rfloor=2 && \lfloor -1 \rfloor=-1\\ \lfloor 2.5 \rfloor=2 && \lfloor -1.5\rfloor=-2\\ \lfloor 2.9 \rfloor=2 && \lfloor -1.1 \rfloor=-2\\ \lfloor 2.99999999 \rfloor=2 && \lfloor -1.00000001 \rfloor=-2\\ \end{align}$$

The ceiling function, $f(x)=\lceil x \rceil$ (also written as $\text{ceil}(x)$), takes a real number $x$ and returns the least integer greater than or equal to $x$.

$$\begin{align} \lceil 3 \rceil=3 && \lceil -4 \rceil=-4\\ \lceil 3.5 \rceil=4 && \lceil -4.5 \rceil=-4\\ \lceil 3.1 \rceil=4 && \lceil -4.9 \rceil=-4\\ \lceil 3.00000001 \rceil=4 && \lceil -4.99999999 \rceil=-4\\ \end{align}$$

Sign function

The sign function takes a real number $x$ and returns $1$ if $x$ is positive, $-1$ if $x$ is negative, and $0$ if $x$ is $0$. We can write it as a piecewise function like so: $$f(x)=\begin{cases} 1 & x>0 \\ -1 & x<0 \\ 0 & x=0 \end{cases}$$

Nearest integer function

The nearest integer function takes a real number and returns the nearest integer. There are multiple different ways to address real numbers equidistant between integers such as $0.5, 2.5,$ and $-1.5$; within computer science, these numbers are usually rounded to the nearest even integer, although they are commonly rounded up as well.

(Video) Heaviside Function

by PatrickJMT

This video by patrickJMT explores the Heaviside function, defined via:

$$H(x)=\begin{cases} 0 & x<0 \\ 1 & x\geq 0 \\ \end{cases}$$

Step Functions

Our next type of piecewise function is the step function! A step function looks exactly like it sounds: a series of steps!

The following is an example of a step function: