Question: The greatest integer function, $f(x) = [x]$, $0 < x < 3$ is not differentiable at how many points?

Step 1 (Formula / Governing Law & Given Data): [0.25 Mark]

The greatest integer function $f(x) = [x]$ is defined as the greatest integer less than or equal to $x$. A function $f(x)$ is discontinuous at all integral points $x = n$ (where $n \in \mathbb{Z}$). Since differentiability implies continuity, if a function is discontinuous at a point, it is necessarily non-differentiable at that point.

Step 2 (Step-by-Step Mathematical Working): [0.5 Mark]

• The given interval is $0 < x < 3$.
• The integers lying within this open interval $(0, 3)$ are $x = 1$ and $x = 2$.
• At $x = 1$:
  $\lim_{x \to 1^-} [x] = 0$ and $\lim_{x \to 1^+} [x] = 1$. Since LHL $\neq$ RHL, $f(x)$ is discontinuous at $x = 1$.
• At $x = 2$:
  $\lim_{x \to 2^-} [x] = 1$ and $\lim_{x \to 2^+} [x] = 2$. Since LHL $\neq$ RHL, $f(x)$ is discontinuous at $x = 2$.
• Because the function is discontinuous at $x = 1$ and $x = 2$, it is non-differentiable at these two points.

Step 3 (Final Concluded Answer): [0.25 Mark]

The function $f(x) = [x]$ is not differentiable at two points ($x=1$ and $x=2$).

Correct Option: (B) At only two points

💡 EL10 Concept Insight: Imagine the graph of the greatest integer function as a set of stair-steps; because there is a "jump" or a break in the line every time you hit a whole number, you can't draw a smooth tangent line there, which is why it's not differentiable at those specific spots.