If f : R \to R is a function defined by f(x)= [x - 1] \cos ( 2x - 1 \over 2 )\pi , where [.] denotes the greatest integer function, then f is :

discontinuous at all integral values of x except at x = 1

Limits Continuity and Differentiability PYQ — JEE Main 2021 · 24 Feb · Shift 1 · Q24 with Solution

View written solutionFree

Correct answer: A

We are given

f(x)=[x-1]\cos\left(\frac{2x-1}{2}\pi\right),

where $[\,\cdot\,]$ is the greatest integer function.

We need to determine where $f$ is continuous.

1. Simplify the trigonometric factor

Observe that

\frac{2x-1}{2}\pi=\left(x-\frac12\right)\pi.

\cos\left(\frac{2x-1}{2}\pi\right)=\cos\left(\left(x-\frac12\right)\pi\right).

Using

\cos\left(x\pi-\frac\pi2\right)=\sin(x\pi),

we get

\cos\left(\left(x-\frac12\right)\pi\right)=\sin(\pi x).

Hence

f(x)=[x-1]\sin(\pi x).

2. Identify possible points of discontinuity

$\sin(\pi x)$ is continuous for all real $x$ .
$[x-1]$ is discontinuous only when $x-1$ is an integer, i.e. when $x$ is an integer.

Therefore, the only possible discontinuities of $f$ are at integer values of $x$ . For non-integer $x$ , both factors are continuous, so $f$ is continuous there.

3. Check continuity at an integer point

Let $x=n$ , where $n\in\mathbb Z$ . We examine left and right behavior.

3.1 Value at $x=n$

Since

\sin(\pi n)=0,

we have

f(n)=[n-1]\sin(\pi n)=0.

3.2 Left-hand limit as $x\to n^-$

For $x\to n^-$ ,

[x-1]=n-2

(because $x-1\to n-1^-$ ). Thus

\lim_{x\to n^-} f(x)=\lim_{x\to n^-} (n-2)\sin(\pi x)=(n-2)\cdot 0=0.

3.3 Right-hand limit as $x\to n^+$

For $x\to n^+$ ,

[x-1]=n-1.

Thus

\lim_{x\to n^+} f(x)=\lim_{x\to n^+} (n-1)\sin(\pi x)=(n-1)\cdot 0=0.

So,

\lim_{x\to n^-} f(x)=\lim_{x\to n^+} f(x)=f(n)=0.

Hence $f$ is continuous at every integer $n$ .

4. Final conclusion

At non-integer $x$ : continuous.
At integer $x$ : also continuous.

Therefore, $f$ is continuous for every real $x$ .

So the correct option is:

\boxed{\text{A}}

5. Comparison with stored correct answer

Stored correct answer: A

Our derived answer is also A, so they agree.

More from Limits Continuity and Differentiability

View written solutionFree

Correct answer: A

We are given

f(x)=[x-1]\cos\left(\frac{2x-1}{2}\pi\right),

where $[\,\cdot\,]$ is the greatest integer function.

We need to determine where $f$ is continuous.

1. Simplify the trigonometric factor

Observe that

\frac{2x-1}{2}\pi=\left(x-\frac12\right)\pi.

\cos\left(\frac{2x-1}{2}\pi\right)=\cos\left(\left(x-\frac12\right)\pi\right).

Using

\cos\left(x\pi-\frac\pi2\right)=\sin(x\pi),

we get

\cos\left(\left(x-\frac12\right)\pi\right)=\sin(\pi x).

Hence

f(x)=[x-1]\sin(\pi x).

2. Identify possible points of discontinuity

$\sin(\pi x)$ is continuous for all real $x$ .
$[x-1]$ is discontinuous only when $x-1$ is an integer, i.e. when $x$ is an integer.

Therefore, the only possible discontinuities of $f$ are at integer values of $x$ . For non-integer $x$ , both factors are continuous, so $f$ is continuous there.

3. Check continuity at an integer point

Let $x=n$ , where $n\in\mathbb Z$ . We examine left and right behavior.

3.1 Value at $x=n$

Since

\sin(\pi n)=0,

we have

f(n)=[n-1]\sin(\pi n)=0.

3.2 Left-hand limit as $x\to n^-$

For $x\to n^-$ ,

[x-1]=n-2

(because $x-1\to n-1^-$ ). Thus

\lim_{x\to n^-} f(x)=\lim_{x\to n^-} (n-2)\sin(\pi x)=(n-2)\cdot 0=0.

3.3 Right-hand limit as $x\to n^+$

For $x\to n^+$ ,

[x-1]=n-1.

Thus

\lim_{x\to n^+} f(x)=\lim_{x\to n^+} (n-1)\sin(\pi x)=(n-1)\cdot 0=0.

So,

\lim_{x\to n^-} f(x)=\lim_{x\to n^+} f(x)=f(n)=0.

Hence $f$ is continuous at every integer $n$ .

4. Final conclusion

At non-integer $x$ : continuous.
At integer $x$ : also continuous.

Therefore, $f$ is continuous for every real $x$ .

So the correct option is:

\boxed{\text{A}}

5. Comparison with stored correct answer

Stored correct answer: A

Our derived answer is also A, so they agree.

Limits Continuity and Differentiability question

Limits Continuity and Differentiability question

1. Simplify the trigonometric factor

2. Identify possible points of discontinuity

3. Check continuity at an integer point

3.1 Value at $x=n$

3.2 Left-hand limit as $x\to n^-$

3.3 Right-hand limit as $x\to n^+$

4. Final conclusion

5. Comparison with stored correct answer

More from Limits Continuity and Differentiability

Limits Continuity and Differentiability question

Limits Continuity and Differentiability question

1. Simplify the trigonometric factor

2. Identify possible points of discontinuity

3. Check continuity at an integer point

3.1 Value at $x=n$

3.2 Left-hand limit as $x\to n^-$

3.3 Right-hand limit as $x\to n^+$

4. Final conclusion

5. Comparison with stored correct answer

More from Limits Continuity and Differentiability

1. Simplify the trigonometric factor

2. Identify possible points of discontinuity

3. Check continuity at an integer point

3.1 Value at x=nx=nx=n

3.2 Left-hand limit as x→n−x\to n^-x→n−

3.3 Right-hand limit as x→n+x\to n^+x→n+

4. Final conclusion

5. Comparison with stored correct answer

More from Limits Continuity and Differentiability

1. Simplify the trigonometric factor

2. Identify possible points of discontinuity

3. Check continuity at an integer point

3.1 Value at x=nx=nx=n

3.2 Left-hand limit as x→n−x\to n^-x→n−

3.3 Right-hand limit as x→n+x\to n^+x→n+

4. Final conclusion

5. Comparison with stored correct answer

More from Limits Continuity and Differentiability

3.1 Value at $x=n$

3.2 Left-hand limit as $x\to n^-$

3.3 Right-hand limit as $x\to n^+$

3.1 Value at $x=n$

3.2 Left-hand limit as $x\to n^-$

3.3 Right-hand limit as $x\to n^+$