Question

Let $f: [-1, 2] \rightarrow \mathbb{R}$ be given by $f(x) = 2x^2 + x + [x^2] - [x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is:

• A. 6
• B. 3
• C. 4
• D. 5

Answer 1

Answer: (C)

4

Answer 2

Given $f(x) = 2x^2 + x + \lfloor x^2 \rfloor - \lfloor x \rfloor$, we analyze its continuity. The floor function, $\lfloor x \rfloor$, introduces discontinuities at integer points since it changes its value abruptly.

Step 1: Points of discontinuity from $\lfloor x \rfloor$ and $\lfloor x^2 \rfloor$:

1. The term $\lfloor x \rfloor$ is discontinuous at all integer values of $x$ within the interval $[-1, 2]$, i.e., at $x = -1, 0, 1, 2$. 2. The term $\lfloor x^2 \rfloor$ introduces discontinuities at points where $x^2$ crosses an integer value. On solving: - For $x^2 = 0, 1, 4$: - $x = -2, -1, 0, 1, 2$ (within $[-1, 2]$, only $x = -1, 0, 1$).

Step 2: Combine discontinuities: Overall, the combined discontinuities occur at the union of these points:

$\text{Points: } x = -1, 0, 1, 2.$

- Since these four points involve jumps in either $\lfloor x \rfloor$ or $\lfloor x^2 \rfloor$, $f(x)$ is discontinuous at exactly 4 points.

Thus, the total number of discontinuities is 4.