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.
Can you write the function where is approximated to ;i.e.:
. (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 .
Summing the right-hand inequalities up, we get
and the conclusion follows.
Solve the equation , where is the integer part of the real number .
Now first analyze the case in which there is no solution,there are two such cases:
For there is no solution.
there is a solution for only.
Solving for these values of we get
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
Okay; now : PRACTICE.
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.