step function


last updated: 20051113 
Using only horizontal line segments ^{1)}, the step
or staircase function is formed.
Every discrete function can be seen
as such a step function.
Another
function that is
sometimes called the step function, is the entier
function int(x) or [x]. Now the steps are of equal length: the function
gives for each x the first lower natural
number. The entier function can be used to form the triangle curve.
For esthetic purposes the horizontal line segments can be connect by vertical
segments.
This conflicts with on the properties of a function, being that for each value
of x there is not more than one function value.
The function of Heaviside H_{a}()
has only one step: until x = a it has a value zero, above the value one. The function describes the velocity
of an object as result of a sudden force, during a infinitely small time interval. H is
the abbreviated notation for H_{0}, what is in fact the same as the unit step function. At the place of the step the value
of the function can be set to ½.
This is in fact the same function as the sign function or signum
function sgn(x) = x / x, which gives the sign of a number.
The rectangle function P(x)
is equal to 0 outside an interval, and equal to 1 inside. It can be expressed as
the difference of two functions of Heaviside.
The curve is also called pulse function, gate function, or window
function.
The boxcar function B_{c}(a,b) is a
variant on this theme: value c for a <= x <= b and 0 otherwise.
The Fourier transform of the function is the sinc
function.
When taking only integer values for x, we get the discrete
curves.
notes
1) By the way, because
it's the case of a function, vertical line segments are not admitted. 