Question

where is the greatest integer function ( f(x)=lfloor x
floor ) not differentiable? integers rational numbers negative numbers 0 ( mathbb{r} ) find a formula for ( f ) where it is defined. ( f(x)=0 ) sketch the graph of ( f ) on the given interval (-2,2).

Explanation:

Step1: Recall the definition of greatest - integer function

The greatest - integer function $f(x)=[x]$ gives the greatest integer less than or equal to $x$. For example, $[2.3]=2$, $[-1.5]= - 2$.

Step2: Analyze the non - differentiability

The graph of $y = [x]$ has a jump discontinuity at every integer value of $x$. At these points, the left - hand limit and the right - hand limit of the difference quotient are not equal. So, $f(x)=[x]$ is not differentiable at integers.

Step3: Analyze the differentiability between integers

For any open interval $(n,n + 1)$ where $n\in\mathbb{Z}$, $f(x)=[x]=n$ (a constant function). The derivative of a constant function $y = C$ (where $C$ is a constant) is $y'=0$. So, $f'(x)=0$ for $x
otin\mathbb{Z}$.

Answer:

1. The greatest - integer function $f(x)=[x]$ is not differentiable at integers.
2. $f'(x)=0$ for $x

otin\mathbb{Z}$

1. To sketch $y = f'(x)$ on the interval $[-2,2]$:

• The function $y = f'(x)$ is $0$ for all $x\in[-2,2]$ except at $x=-2,-1,0,1,2$. At these integer points, the function $f'(x)$ is not defined. So, we draw a horizontal line $y = 0$ on the open intervals $(-2,-1),(-1,0),(0,1),(1,2)$ with open - circles at the endpoints $x=-2,-1,0,1,2$.