Let f : [0, 3] → R be defined by f(x) = min{x - [x], 1 + [x] - x} where [x] is the greatest integer

Q: Wait, why is the function continuous everywhere? Doesn't the Greatest Integer Function usually cause jumps?

That is a fantastic observation! You are right that $[x]$ itself is discontinuous at every integer. However, in our function $f(x) = \min(\{x\}, 1 - \{x\})$, the discontinuities of $[x]$ are 'hidden' inside the fractional part. The fractional part function $\{x\}$ is continuous from the right and has jumps at integers, but notice that as $x$ approaches an integer $n$ from the left, $\{x\} \to 1$, and as $x$ approaches $n$ from the right, $\{x\} \to 0$. When you plug these into our specific $\min$ function, the values align perfectly at $0$, smoothing out the jump. It is a beautiful example of how a combination of discontinuous functions can result in a continuous one!

Q: Why are the valleys at $x=1$ and $x=2$ considered non-differentiable?

Great question! Differentiability requires a smooth transition. At the valleys (the integers), the function switches from the line $y = 1 - x$ (or $1 - \{x\}$) to the line $y = x$ (or $\{x\}$). If you calculate the slope just before $x=1$, you get $-1$. Just after $x=1$, the slope is $+1$. Because the left-hand derivative ($-1$) does not equal the right-hand derivative ($+1$), there is a sharp 'kink' or corner at the valley. Even though the graph is continuous, that sharp change in direction makes it non-differentiable.

Q: Why do we exclude $x=0$ and $x=3$ from the set $Q$?

The question specifically asks for the set $Q$ to contain all $x \in (0, 3)$. This is an open interval notation. In mathematics, when we define an interval with parentheses like $(0, 3)$, we are excluding the endpoints. Even if the function had a sharp corner at $x=0$ or $x=3$, we would not count them because they fall outside the specified range. Always pay close attention to whether the interval is open or closed!

Question

Let $f : [0, 3] 	o \mathbb{R}$ be defined by $f(x) = \min\{x - [x], 1 + [x] - x\}$ where $[x]$ is the greatest integer less than or equal to $x$. Let $P$ denote the set containing all $x \in [0, 3]$ where $f$ is discontinuous, and $Q$ denote the set containing all $x \in (0, 3)$ where $f$ is not differentiable. Then the sum of number of elements in $P$ and $Q$ is equal to

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine standing before a function that looks like a tangled mess of brackets and variables: 
$$f(x) = \min\{x - [x], 1 + [x] - x\}$$

It is natural to feel a surge of intimidation. The Greatest Integer Function, $[x]$, is notorious for its abrupt jumps and stubborn behavior. But in the world of JEE Advanced, we do not fear these functions; we decode them.

### Decoding the Mystery

The first step in our journey is to strip away the complexity. We know that the fractional part of $x$, denoted by $\{x\}$, is defined as $\{x\} = x - [x]$.

If we look at the first term of our function, $x - [x]$, it is exactly $\{x\}$. Now, look at the second term: $1 + [x] - x$.

If we factor out a negative sign, we get $1 - (x - [x])$, which is simply $1 - \{x\}$. Suddenly, the terrifying expression transforms into something remarkably simple:
$$f(x) = \min(\{x\}, 1 - \{x\})$$

We are no longer dealing with integers; we are dealing with the fractional part, which behaves predictably between $0$ and $1$.

### The Geometric Visualization

Let us analyze this in the interval $x \in [0, 1)$. Here, $[x] = 0$, so $\{x\} = x$. Our function becomes:
$$f(x) = \min(x, 1 - x)$$

If you graph $y = x$ and $y = 1 - x$, they intersect at $x = 0.5$. For $x < 0.5$, $x$ is smaller, so $f(x) = x$. For $x > 0.5$, $1 - x$ is smaller, so $f(x) = 1 - x$.

This creates a triangular peak at $x = 0.5$. Because the fractional part function is periodic with a period of $1$, this triangular shape repeats itself. We have created a sawtooth wave where peaks occur at $x = 0.5, 1.5, 2.5$ and valleys occur at the integers $x = 0, 1, 2, 3$.

### The Anatomy of Continuity

Now, let us address the set $P$, the points of discontinuity. As we trace our sawtooth wave from $x = 0$ to $x = 3$, we see a continuous, connected path.

There are no jumps, no holes, and no vertical asymptotes. Every time the function hits a valley at an integer, the value of $\{x\}$ resets from $1$ to $0$, but because our function is the minimum of $\{x\}$ and $1 - \{x\}$, the values meet at $0$.

Thus, the function is continuous everywhere. The set $P$ is empty, and the number of elements $n(P) = 0$.

### The Sharp Truth: Differentiability

Finally, we arrive at the set $Q$, the points of non-differentiability. A function fails to be differentiable wherever it has a 'sharp corner'—a point where the slope changes abruptly.

Looking at our graph, we see these corners at every peak and every valley. At the peaks (e.g., $x = 0.5$), the slope changes from $+1$ to $-1$. At the valleys (e.g., $x = 1$), the slope changes from $-1$ to $+1$.

In both cases, the left-hand derivative does not equal the right-hand derivative. Within the open interval $(0, 3)$, these sharp corners occur at $x = 0.5, 1, 1.5, 2, 2.5$. Counting them up, we find $5$ such points.

### The Final Victory

We have navigated the complexity, visualized the geometry, and applied the definitions of continuity and differentiability. The set $P$ has $0$ elements, and the set $Q$ has $5$ elements.

The sum is $0 + 5 = \mathbf{5}$. You have successfully dismantled a problem that once looked intimidating and turned it into a clear, logical victory.

Enter Numerical Value:

Visualized Solution

Understanding the Function $f (x)$

Recalling the Fractional Part Function

Simplifying the Function

Analyzing the Interval $[0, 1)$

Finding the Intersection Point

Visualizing the Graph

Checking Continuity (Set $P$ )

Identifying Non-Differentiable Points (Set $Q$ )

Analyzing the Peaks

Analyzing the Valleys

Final Counting and Sum

The Sigma Insight: Relationship Between Continuity and Differentiability

Analyzing the Setup

Decoding the Mystery

The Geometric Visualization

The Anatomy of Continuity

The Sharp Truth

The Final Victory

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