Which of the following functions is periodic? (a) f(x) = x - [x] where [x] denotes the largest integ

Q: Wait, $\sin(x)$ is periodic, so why isn't $\sin(\frac{1}{x})$ periodic?

This is a classic trap! Periodicity requires the *entire* function to repeat its values over a fixed interval $T$. While $\sin(\theta)$ is periodic with respect to $\theta$, the argument $\frac{1}{x}$ changes the 'speed' of the oscillation as $x$ changes. Near $x=0$, the frequency becomes infinite, meaning the function oscillates faster and faster. Because the frequency is not constant, there is no fixed $T$ that can make the function repeat itself.

Q: Why can't we just say $f(x) = x \cos x$ is periodic because $\cos x$ is periodic?

Multiplication by $x$ creates a 'variable amplitude.' Think of it this way: $\cos x$ oscillates between $-1$ and $1$. But $x \cos x$ oscillates between $-x$ and $x$. As $x$ increases, the peaks of the wave grow taller and taller. Periodicity requires the *values* to repeat, not just the shape. Since the height of the peaks changes, the function never returns to the same value after a fixed interval.

Q: Is the fractional part function $f(x) = \{x\}$ really periodic? It has those jumps!

Yes, it is! The definition of periodicity does not require the function to be continuous. It only requires $f(x+T) = f(x)$. Even though there is a jump discontinuity at every integer, the pattern of the 'sawtooth' repeats exactly every $T=1$ unit. The jump at $x=1$ is identical to the jump at $x=2$, and so on. It is perfectly periodic.

Question

Which of the following functions is periodic?

Options:

(A) $f(x) = x - [x]$ where $[x]$ denotes the largest integer less than or equal to the real number $x$

(B) $f(x) = \sin \frac{1}{x}$ for $x 
eq 0, f(0) = 0$

(C) $f(x) = x \cos x$

(D) none of these

Sigma Percentile · Accepted Answer

### The Rhythm of Mathematics: Understanding Periodicity

Welcome, future engineers. Today, we are diving into one of the most elegant concepts in calculus: periodicity. In the JEE Advanced, examiners love to test whether you truly understand a definition or if you are just relying on visual intuition.

A function $f(x)$ is periodic if there exists a positive constant $T > 0$ such that $f(x + T) = f(x)$ for all $x$ in the domain. Think of it as a heartbeat—a rhythm that repeats perfectly.

If you take a snapshot of the graph over an interval of length $T$, you can copy and paste it infinitely to the left and right, and the graph will look identical.

### The Fractional Part: The Sawtooth Wave

Let's examine our first candidate: $f(x) = x - [x]$. This is the fractional part function, often denoted as $\{x\}$.

Imagine you are walking along the number line. As you move from $0$ to $1$, the value of $x$ increases, and since the greatest integer $[x]$ is $0$, the function simply traces the line $y=x$. But the moment you hit $1$, the greatest integer jumps to $1$, and the function resets to $0$. This creates a beautiful 'sawtooth' pattern.

To prove it is periodic, we test the definition:
$$f(x+1) = (x+1) - [x+1]$$

Using the fundamental property of the greatest integer function, we know that $[x+1] = [x] + 1$. Substituting this, we get:
$$f(x+1) = x + 1 - ([x] + 1) = x - [x] = f(x)$$

The $+1$ and $-1$ cancel out with surgical precision! This confirms that the function repeats every $T=1$. It is **periodic**.

### The Trap of Non-Linearity: $\sin(\frac{1}{x})$

Now, look at $f(x) = \sin(\frac{1}{x})$. Many students see the word 'sine' and immediately think 'periodic.' But pause.

The sine function is periodic with respect to its *argument*. If the argument is linear, like $\sin(x)$ or $\sin(2x)$, it is periodic. But here, the argument is $\frac{1}{x}$.

As $x$ approaches $0$, the value of $\frac{1}{x}$ shoots toward $\infty$. This means the sine function is forced to oscillate between $-1$ and $1$ infinitely many times in a tiny neighborhood of the origin.

The frequency of oscillation is not constant; it is accelerating. Because the 'rhythm' of the wave changes as you move along the $x$-axis, there is no fixed $T$ that can capture this behavior. It is **non-periodic**.

### The Trap of Variable Amplitude: $x \cos x$

Finally, consider $f(x) = x \cos x$. This is a product of a linear function $x$ and a periodic function $\cos x$.

The $\cos x$ term tries to keep the rhythm, but the $x$ term acts as a 'variable amplitude.' Imagine a wave that grows taller as it travels. At $x=0$, the height is $0$. At $x=2\pi$, the height is $2\pi$. The peaks are getting higher and higher.

If we test the definition:
$$f(x + 2\pi) = (x + 2\pi) \cos(x + 2\pi) = (x + 2\pi) \cos x = x \cos x + 2\pi \cos x$$

This is clearly not equal to $f(x)$. The extra term $2\pi \cos x$ proves that the function does not return to its original value. It is **non-periodic**.

### The Verdict

We have dissected these functions with mathematical rigor. We saw that periodicity is not just about seeing a wave; it is about the algebraic identity $f(x+T) = f(x)$.

Only the fractional part function satisfies this condition. Keep this mindset—always verify with algebra, never trust your eyes alone. You are doing great.

Animated Solution for Mathematics - Functions: Which of the following functions is periodic?

Select Answer:

Visualized Solution

What is a Periodic Function?

Analyzing $f (x) = x - [x]$

Mathematical Proof for $f (x) = {x}$

Analyzing $f (x) = sin (\frac{1}{x})$

Infinite Oscillations Near the Origin

Analyzing $f (x) = x cos x$

Amplitude Growth and Non-Periodicity

Final Verdict

The Sigma Insight: Periodic Functions

The Rhythm of Mathematics

The Fractional Part

The Trap of Non-Linearity: $sin (\frac{1}{x})$

The Trap of Variable Amplitude: $x cos x$

The Verdict