Prove that the greatest integer function `[x]` is continuous at all points except at integer points.

The function f(x)=x-[x] , where [⋅] denotes the greatest integer function is (a) continuous everywhere (b) continuous at integer points only (c) continuous at non-integer points only (d) differentiable everywhere

f(x)=[sinx] where [.] denotest the greatest integer function is continuous at:

Let f(x)={{:(,1+x,"if "x lt 0),(,1+[x]+sin x,0 le x lt pi//2),(,3,x ge pi//2):} Statement-1: F is a continuous on R-[1] Statement-2: The greatest integer function is discontinuous at every integer point.

Let f(x)={8^(1/x),x<0a[x],a in R-{0},xgeq0, (where [.] denotes the greatest integer function). Then (a) f(x) is Continuous only at a finite number of points (b)Discontinuous at a finite number of points. (c)Discontinuous at an infinite number of points. (d)Discontinuous at x=0

Five examples of greatest integer function integer notation are:

Prove that the greatest integer function f:RtoR , given by f(x)=[x] is a many-one function.

The function f(x)=(sin(pi[x-pi]))/(4+[x]^2) , where [] denotes the greatest integer function,(a) is continuous as well as differentiable for all x in R (b) continuous for all x but not differentiable at some x (c) differentiable for all x but not continuous at some x . (d) none of these

The function f(x)=(sin(pi[x-pi]))/(4+[x]^2) , where [] denotes the greatest integer function, (a) is continuous as well as differentiable for all x in R (b) continuous for all x but not differentiable at some x (c) differentiable for all x but not continuous at some x . (d) none of these

Prove that the greatest integer function f: RR rarr RR , given by f(x)=[x] , is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Prove that the function f(x)=x^n is continuous at x = n , where n is a positive integer.