Properties of Greatest Integer Function (ix) `[x]ltk implies xltk; where k epsilon Z` (x)`[x+y]=[x]+[y+x-[x]]` for all `x;yepsilon R`

Properties of Greatest Integer Function (vi)[x]>=k rArr x>=k; whereke Z(vii)[x] k rArr x>(k+1) ; where k varepsilon Z

Properties of smallest integer function property (i)[-n]=-[n]where n epsilon N (ii) [-x]=-[x]+1; where xepsilon R-Z (iii) [x+n]=[x]+n where x epsilon R-Z and n epsilon Z (iv) [x]+[-x]={1; x notinZ ; 0 ; xinZ} (v) [x]-[-x]={2[x]-1; xnotinZ ; 2[x] ; if xinZ

Sketch the graph of the following function: (i) f(x)=[Ix-1].] denotes the greatest integer function (ii) f(x)=x+[x.] denotes the greatest integer function.? eli- f(x)=sin|?[x].[.] denotes the greatest integer function.2

Find the left hand and right hand limits of greatest integer function f(x)=[x]= greatest integer less than or equal to x, at x=k, where k is an integer.

Find all the points of discontinuity of the greatest integer function defined by f(x)=[x], where [x] denotes the greatest integer less than or equal to x.

Show that f(x)=[x] , ([] implies greatest integer function ) is increasing function on R.

Let f:" "R->R be the Signum Function defined as f(x)={1,""""""""""""x >0 0,""""""""x=0-1,""""""""x R be the Greatest Integer Function given by g(x)" "=" "[x] , where [x] is greatest integer less than or equal to x. Then does fo