# The Floor Function/Box Function

Definition
We define the floor function as : greatest integer less than or equal to . e.g., .Graph of the floor function
The function is shown in the image (1st quadrant only)(see the attachment)

This function is discontinuous and indifferentiable at each and every points.

Some other definitions
We define “curvy x” as:
We define the ceiling function as: the least integer greater than or equal to ; ie least integer greater than or equal to . For example; .

Hence it is easy to notice that for

A challenge
Can you write the function where is approximated to ;i.e.:

Common concepts

(Proof:
Let .

Hence
Also
Comparing (1)&(2) yields the desired result. )

Examples
. (alex2008) Prove that .
(solution by Potla)
See this url for proof. (refer to post # 126, 127)

For every ; find the largest , such that:

Hint: Define . Consider different values of to get a recurrence relation. Then set .

.(alex2008 – inequation) Solve the following inequation :

(solution by omegatheo)

.

(alex2008; concepts of functions needed) Find all the functions such that
(solution by Farenhajt)
Putting we get

Putting , we get

Putting where and , we get

Putting in we get

Now becomes

Hence

(alex2008)Let be . Show that :

(solution by Farenhajt)
Since , we get , hence

Summing the right-hand inequalities up, we get

and the conclusion follows.

Solve the equation , where is the integer part of the real number .
Solution (makar)
Let
and
and
Now first analyze the case in which there is no solution,there are two such cases:
or
For there is no solution.
there is a solution for only.
Solving for these values of we get

References
http://www.artofproblemsolving.com/Foru … p?t=273532
http://www.artofproblemsolving.com/Foru … p?t=273536
http://www.artofproblemsolving.com/Foru … p?t=273069
http://www.artofproblemsolving.com/Foru … p?t=281136

MY APOLOGISES

Okay; now : PRACTICE.
EXERCISES

1. Prove that we have:

2. Prove that we have the following identity:

Where .

( important lemma)The “golden ratio” is defined as . Prove that : ()
(a) or .
(b) 1. (if ) ; and 2.
otherwise.

. If and we have or .

. (inequality; USAMO 1981) Prove that and we have: .

I leave you all (mainly beginners) to digest these first, and then I will post the solutions. If any of you have some solutions, please post it as a comment to this post.