Prove theat the greatest interger function defined by `f(x)= |x|, 0 lt x lt 3` is not differentiable at x=1 and x= 2.

The function f(x) = {{:(2, x lt=1),(x , x gt 1 ):} is not differentiable at ………..

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

Find the intervals in which the function f given by f(x) =sin x +cos x , lt=x lt=2pi is increasing or decreasing.

The function f is defined by f(x) = {{:(1-x, x lt 0) ,( 1, x=0),( x+1,x gt 0):} draw the graph of f(x) .

Discuss the continuity of the function defined by {{:(x+2," if "x lt 0),(-x+2," if "x gt 0):}

Draw the graph of the function f(x ) = {{:(1+x , -1 le x le 0),( 1-x, 0 lt x le 1):}

Given f(x) is a periodic function with period 2 and it is defined as " "f(x) = {{:([cos""(pix)/(2)]+1",",,0lt x lt 1), (2-x",",,1 le x lt 2):} Here [*] represents the greatest integer le x . If f(0)=1 , then draw the graph of the function for x in [-2, 2] .

f(x)={[sinpix],0lt=x<1sgn(x-5/4)x{x-2/3},1lt=xlt=2 where [.] denotes the greatest integer function and {.} represents the fractional part function. At what points should the continuity be checked? Hence, find the points of discontinuity.