#### To determine

**To find:** The points where the greatest integer function f(x)=〚x〛 is not

differentiable and then to find the formula for f′ and sketch the graph.

#### Answer

The greatest integer function is not differentiable at x∈ℤ

The formula for the derivative function is f′(x)={0if x∈ℤundefinedif x∈ℤ.

#### Explanation

**Result Used:**

The greatest integer function is defined as follows,

f(x)={y if x∈ℤ:y is nearest interger less than xx if x∈ℤ

**Calculation:**

Obtain the points where the greatest integer function f(x)=〚x〛 is not differentiable.

The derivative of the greatest integer function is calculated as follows,

**Case 1:** *x* is not an integer.

f′(x)=limh→0f(x+h)−f(x)h=limh→0〚x+h〛−〚x〛h

Since *x* is not an integer, 〚x+h〛=〚x〛.

f′(x)=limh→0〚x〛−〚x〛h=limh→00h=0

Thus, f′(x)=0

**Case 2:** *x* is an integer.

The left hand derivative of the function is computed as follows,

f′−(x)=limh→0−f(x+h)−f(x)h=limh→0−〚x+h〛−〚x〛h

Since *h* approaches 0 from left, x+h<x.

This implies that, 〚x+h〛=x−1.

limh→0−〚x+h〛−〚x〛h=limh→0−x−1−xh=−1limh→0−h=−10

Thus, the left hand derivative of the function does not exist. This follows that, f′(x) does not exist.

Therefore, the greatest integer function is not differentiable at the integer points.

By case (1) and (2), it is defined that the formula for the derivative is f′(x)={0if x∈ℤundefinedif x∈ℤ.

**Graph:**

Use the above information and trace the graph of f′ as shown below in Figure 1,

From the Figure 1, it is clear that the greatest integer function is discontinuous at integer points.