Question

The greatest integer function, \(f(x) = [x]\), \(0 < x < 3\) 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 greatest integer function "jumps" at every integer.
If the interval is closed, check the endpoints carefully, but for open intervals, just count the integers inside.
Non-differentiability also occurs at sharp corners, but for \([x]\), the primary cause is discontinuity.

Answer 1

The Correct Option is B

Concept:
• The greatest integer function \(f(x) = [x]\) is discontinuous at every integer value of \(x\).
• Differentiability implies continuity. Therefore, if a function is discontinuous at a point, it cannot be differentiable at that point.
• In an interval \((a, b)\), we count the number of integers contained within the interval.

Step 1: Identify the points of discontinuity in the given interval
The function is defined for \(x \in (0, 3)\).
The integers within this open interval are \(x = 1\) and \(x = 2\).

Step 2: Analyze differentiability at integer points
At \(x = 1\):
Left Hand Limit (LHL) \(= \lim_{x \to 1^-} [x] = 0\).
Right Hand Limit (RHL) \(= \lim_{x \to 1^+} [x] = 1\).
Since LHL \(\neq\) RHL, the function is discontinuous at \(x = 1\).
At \(x = 2\):
LHL \(= \lim_{x \to 2^-} [x] = 1\).
RHL \(= \lim_{x \to 2^+} [x] = 2\).
Since LHL \(\neq\) RHL, the function is discontinuous at \(x = 2\).

Step 3: Determine the total number of points of non-differentiability
As shown above, the function is discontinuous at \(x = 1\) and \(x = 2\).
Since a function is not differentiable where it is discontinuous, it is not differentiable at \(x = 1\) and \(x = 2\).
The total number of such points in the given interval is 2.