In mathematics, the function that takes a real number as input and
returns its integer part is called the greatest integer
function. The notation used is 
and the formal definition is that
is the largest integer n satisfying 
. Another common
name for this function is the floor function, 
, and
that is the name used by Maple. See the examples below.
> floor(21/4);
> floor(13.543);
> plot(floor(x),x=0..4);
> plot(floor(x),x=0..10);
Next, consider the plot generated by the following Maple command.
> plot(x-floor(x),x=0..4);
To help you understand the plot, recall that if 
, then
so 
. Now, if 
then 
so 
. In fact,
we could describe 
with the following formula
There is one thing to watch out for, though. Compare the plots generated by the following commands.
> plot(x-floor(x),x=0..8);
> plot(x-floor(x),x=0..8,numpoints=1000);
The first plot generated is not an accurate plot of 
. You can
tell this because the peaks are not at the same height, as they should
be. The second command uses the numpoints parameter to tell
Maple to plot more points. There is nothing special about the value
1000, however, you should just increase the value of this parameter
until you get a plot that looks right.
Now, suppose that L is a positive real number and consider the following expression.
You should be able to convince yourself that the following formula holds.
You can try the following Maple commands if you need more help
visualizing how 
behaves.
> plot(x-2*floor(x/2),x=0..4);
> plot(x-6*floor(x/6),x=0..24,numpoints=250);
Notice how the value of L affects not only the interval at which the graph repeats, but also the maximum value attained.
Finally, suppose we have a function 
and we define a new function
by
After a little thought, you should be able to convince yourself that
for any value of x. This means that the function 
only depends
on the values of 
for 
. In fact the graph of
for 
consists of copies of the graph of 
for 
translated to the right by multiples of L.
As an example, consider the function 
. Try the following
Maple commands.
> f := x -> x^2+1;
> g := x -> f(x-floor(x));
> plot(g(x),x=0..5,numpoints=200);
> h := x -> f(x-2*floor(x/2));
> plot(h(x),x=0..5);
> h3 := x -> f(x-3*floor(x/3));
> plot(h3(x),x=0..6);