Question

The greatest integer function, \(f(x) = [x]\), evaluated across the open domain boundary \(0 < x < 3\), is not differentiable at how many points?

• A. At only one point
• B. At only two points
• C. At no point
• D. At three points

Hint:

The step function \(f(x) = [x]\) is always discontinuous and non-differentiable at every single integer point \(x = n \in \mathbb{Z}\). Simply count how many whole numbers fall inside the specified range.

Answer 1

The Correct Option is B

Concept: The greatest integer function, denoted as \([x]\), outputs the greatest integer less than or equal to \(x\).
• A basic mathematical theorem states that for any function to be differentiable at a given point, it must be continuous at that point. If a function is discontinuous at a point, it cannot be differentiable there.
• The greatest integer function \([x]\) undergoes a step-jump discontinuity at every integer value because its left-hand limit and right-hand limit do not match.

Step 1: Identify all integer points within the given domain
The problem specifies the domain interval as: \[ 0 < x < 3 \quad \Rightarrow \quad x \in (0, 3) \] Let us locate all integers that lie strictly within this open interval. The integers contained inside \((0, 3)\) are: \[ x = 1 \quad \text{and} \quad x = 2 \] Note that the endpoint values \(x = 0\) and \(x = 3\) are excluded from this open domain.

Step 2: Analyze continuity at these integer values
Let us look at \(x = 1\):
• Left-Hand Limit (\(\text{LHL}\)): \(\lim_{x \to 1^-} [x] = 0\)
• Right-Hand Limit (\(\text{RHL}\)): \(\lim_{x \to 1^+} [x] = 1\) Since \(\text{LHL} \neq \text{RHL}\), the function is discontinuous at \(x = 1\), which means it is not differentiable at \(x = 1\). Let us look at \(x = 2\):
• Left-Hand Limit (\(\text{LHL}\)): \(\lim_{x \to 2^-} [x] = 1\)
• Right-Hand Limit (\(\text{RHL}\)): \(\lim_{x \to 2^+} [x] = 2\) Since \(\text{LHL} \neq \text{RHL}\), the function is discontinuous at \(x = 2\), which means it is not differentiable at \(x = 2\).

Step 3: Count the total number of non-differentiable points
For all non-integer values inside \((0, 3)\), the function is constant, meaning its derivative exists and equals $0$. Thus, the points of non-differentiability are exactly two: \(x = 1\) and \(x = 2\). This matches choice option (B).