please tell me the basic concept regarding the properties of [x] like

[x+y] = [x] +[y+x-{x}]

[x]+[x+1/n]+[x+2/n]+....+ [x+n-1/n] = [nx]

etc also tell me the applications of this function

also tell me the application of smallest integer function

The function [x] gives the integer immediately left of x.

It can also be expressed as x-{x}

[x]+[x+1/n]+[x+2/n]+....+ [x+n-1/n] = [nx]

because r/n<1 for   r<n-1

so  [x+r/n] = [x]

where r<n

hence the result follows

is it so prathab?    What abt if x = 0.9?  and n = 5?

and taking x to be an integer is the simplest case :D

In any case that formula holds for all x

cant you show how far you have tried or have you tried it all all?

you should atleast show till where you have tried Huh?