Let f(x) = x^2 - x + x, where x R and | t | denotes the greatest… — Solution | JEE Mathematics

MCQMediumJEE 2023Continuity

Let $f(x) = \lfloor x^2 - x \rfloor + \lfloor x \rfloor$ , where $x \in \mathbb{R}$ and $| t |$ denotes the greatest integer less than or equal to $t$ . Then, $f$ is:

A
Not continuous at $x = 0$ and at $x = 1$
B
Continuous at $x = 0$ and at $x = 1$
C
Continuous at $x = 1$ , but not continuous at $x = 0$
D
Continuous at $x = 0$ , but not continuous at $x = 1$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given: $f(x) = \lfloor x^2 - x \rfloor + \lfloor x \rfloor$ .

Find: Whether $f$ is continuous at $x = 0$ and $x = 1$ .

At $x = 0$ ,

f(0) = \lfloor 0^2 - 0 \rfloor + \lfloor 0 \rfloor = \lfloor 0 \rfloor + \lfloor 0 \rfloor = 0

As $x \to 0^-$ and $x \to 0^+$ , the function approaches the same value. Hence, $f$ is continuous at $x = 0$ .

At $x = 1$ ,

f(1) = \lfloor 1^2 - 1 \rfloor + \lfloor 1 \rfloor = \lfloor 0 \rfloor + \lfloor 1 \rfloor = 0 + 1 = 1

Now check the limits from both sides:

\lim_{x \to 1^-} f(x) = 0

and

\lim_{x \to 1^+} f(x) = 0

The left-hand limit and right-hand limit are equal, but this common limit is not equal to $f(1) = 1$ . Therefore, $f$ is not continuous at $x = 1$ .

So, $f$ is continuous at $x = 0$ , but not continuous at $x = 1$ . The correct option is D.

Common mistakes

Confusing the floor function with the modulus function. Here $| t |$ is explicitly defined as the greatest integer less than or equal to $t$ , so it means $\lfloor t \rfloor$ , not absolute value. Always use the given definition.
Checking only the function value and not the one-sided limits. Continuity at a point requires left-hand limit, right-hand limit, and function value to be equal. At $x = 1$ , the limits are equal to $0$ but $f(1) = 1$ .
Assuming floor functions are discontinuous at every integer input without examining the full expression. The combination $\lfloor x^2 - x \rfloor + \lfloor x \rfloor$ can still be continuous at some points such as $x = 0$ , so evaluate the actual behavior near the point.

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