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

The Correct Option is C

To find the number of points where \( f(x) = 2x^2 + x + [x^2] - [x] \) is discontinuous, we examine the greatest integer function \([t]\). Discontinuities for such functions typically occur at integer values where \([x]\) or \([x^2]\) change abruptly.

We will analyze the domain \([-1, 2]\). The critical points where the floor function might cause a discontinuity are:

1. Integer values for \([x]\): \(-1, 0, 1, 2\).
2. Values where \(x^2\) becomes an integer: In \([-1, 2]\), \(x^2 \in \{0, 1, 4\}\). This corresponds to \(x \in \{0, \pm1, \pm2\}\). Within the specified domain, the relevant points are \(-1, 0, 1, 2\).

We evaluate the function's behavior at these critical points:

1. \(x = -1\): At \(x=-1\), \([x] = -1\) and \([x^2] = 1\). The transition at \(-1\) within the domain indicates a discontinuity due to the nature of the floor function.
2. \(x = 0\): The function value is \(2(0)^2 + 0 + [0^2] - [0] = 0\). The jump in \([x]\) at \(0\) is a potential point of discontinuity.
3. \(x = 1\): Both \([x]\) and \([x^2]\) experience changes at \(x=1\), indicating a discontinuity.
4. \(x = 2\): The transition of \([x]\) at the boundary of the domain suggests a potential discontinuity.

The identified points of discontinuity are \(-1, 0, 1, 2\). Therefore, the function is discontinuous at 4 points.