Problem 158 - A Complete Resource Book in Mathematics for JEE Main

Question

A function is said to be bounded if its range is bounded, otherwise, it is unbounded. Thus, a function \(f(x)\) is bounded in the domain \(D\), if there exist two real numbers \(k\) and \(K\) such that $$ k \leq f(x) \leq K \text { for all } x \in D $$ Again, the bounds of the range of a bounded function are called the bounds of the function. Let \(f: X \rightarrow Y\) (i.e., \(f\) is a function whose domain is \(X\) and range \(f(X) \subseteq Y\) the co-domain). \(f\) is called a monotonically increasing function if \(x_{1}, x_{2} \in X\) with \(x_{1}

Step-by-Step Solution

Answer

(B) \( m \)

Step 1: Identify the function behavior

The function \( f(x) \) is defined piecewise: it is 0 when \( x \) is an integer and 1 otherwise. This means the function is 1 for almost every point in any interval, except at integers where \( f(x) = 0 \).

Step 2: Determine the discontinuity points

The function \( f(x) \) is discontinuous at every integer in the interval \([0, m]\). However, the set of points where the function is discontinuous (the integers) is finite within any bounded interval.

Step 3: Check integrability conditions

Even though the function is discontinuous at integer points, these discontinuity points are finite in the interval \([0, m]\), which means the function satisfies a condition for integrability: a bounded function with finitely many discontinuities is integrable.

Step 4: Evaluate the integral

Since \( f(x) = 1 \) almost everywhere in the interval \([0, m]\), except at finitely many points, the integral evaluates mainly as an integral of the constant function 1. Thus, the integral \( \int_{0}^{m} f(x) dx = m \), close to tallying up lengths of sub-intervals where \( f(x) = 1 \).

Key Concepts

Bounded Function

A bounded function is one where its range is limited within specific numerical bounds. Imagine it as a set of guardrails that keep the function's output from veering too far off course. For a function \( f(x) \) to be bounded over a domain \( D \), there must be two real numbers \( k \) and \( K \) such that:\[k \leq f(x) \leq K \quad \text{for all}\ x \in D\]Meaning, no matter the input \( x \) within the domain, the output \( f(x) \) will always lie between \( k \) and \( K \).
In terms of a very simple analogy, think of it like a roller coaster that never goes below the ground or above its highest tracks. Its entire path is capped. Understanding whether a function is bounded is crucial for determining whether other properties, like integrability, hold.

Discontinuity Points

Discontinuity points refer to where a function abruptly changes its value. You can think of them as interruptions or breaks in the function's path. When looking at a graph, discontinuity might show as a jump from one level to another, or a hole where a function is undefined. For instance, the function \( f(x) = 0 \) if \( x \) is an integer and \( 1 \) otherwise, exhibits discontinuities at every integer point.
These are spots where the beautiful continuous curve of the function is interrupted.
An important principle is that if a bounded function has a finite set of discontinuity points over a specific interval, it retains integrability over that interval. Integrability just means that you can calculate its integral, or the area under its curve, despite these little hiccups along the way.

Monotonically Increasing Function

A function is monotonically increasing when its value never decreases as its input progresses. It's like climbing up a hill; the higher you go, the higher up you are. Mathematically, a function \( f \) is monotonically increasing if, for any two inputs \( x_1 \) and \( x_2 \) within its domain, where \( x_1 < x_2 \), it follows that:\[f(x_1) \leq f(x_2)\]This simply means that if you were to plot the function on a graph, you'd have a non-decreasing curve. No matter where you stop along the domain, the function will never slope downwards.
Monotonically increasing functions have a neat property; if they are bounded and occur over a finite interval, they ensure integrability. It simplifies many things, like calculating areas under curves, since the function doesn't unpredictably dip below its own previous values.