Question

The greatest integer function defined by \( f(x) = [x] \), \( 1 < x < 3 \) is not differentiable at \( x = \):

• (A) 0
• (B) 1
• (C) 2
• (D) 3/2

Answer 1

The Correct Option is C

Step 1: Define the Greatest Integer Function
The greatest integer function, denoted by [x], yields the largest integer less than or equal to x. This is also known as the floor function.

Step 2: Describe the Greatest Integer Function's Behavior
The function remains constant between integers but experiences abrupt changes at integer values. For instance, [1.5] = 1 and [2.3] = 2. At x = 2, the function value transitions from 1 to 2.

Step 3: Evaluate Differentiability and Continuity
Differentiability requires continuity and smoothness at a point. The greatest integer function exhibits jump discontinuities at all integer points, rendering it non-differentiable at these locations.

Step 4: Examine the Interval 1 < x < 3
Within this interval, the function has discontinuities at the integers 2 and 3. Therefore, the points of non-differentiability are x = 2 and x = 3.

Step 5: Conclude
As the question asks for non-differentiability within the open interval 1 < x < 3, the relevant point is x = 2, excluding x = 3.

Final Answer: (C) 2