Question

The greatest integer function is continuous on r true or false

Answer 1

A function is said to be continuous when graph of function doesn't have any break or hole . Mathematically , A function f(x) is continuous at x = a only when,
$\bold{Lim_{x\to a^+}\,f(x)=Lim_{x \to a^-}\,f(x) = f(a) }$

A greatest integer Function is piece-wise function .
f(x) = [x] where [x ] ≤ x

f(x) = 0 when 0 ≤ x < 1
f(x) = 1 when 1 ≤ x < 2
f(x) = 2 when 2 ≤ x < 3
f(x) = -1 when -1 ≤ x < 0
f(x) = -2 when -2 ≤ x < -1
Plot graph with help of above situations .you get graph of f(x) = [x] as shown in figure. What you observed ? Is this continuous function ?
Of course , it is not continuous function. Because graph is breaking each integer point.

Hence, this is false to say " greatest integer Function is continuous on R .