Problem 51 - A Complete Resource Book in Mathematics for JEE Main

Question

The Dirichlet function, defined as \(f(x)=\left\\{\begin{array}{l}1 \text { if } x \text { is rational } \\ 0 \text { if } x \text { is irrational }\end{array}\right.\), is (A) continuous for all real \(x\) (B) continuous only at some values of \(x\) (C) discontinuous for all real \(x\) (D) discontinuous only at some values of \(x\)

Step-by-Step Solution

Answer

(C) discontinuous for all real \(x\)

Step 1: Understand the Dirichlet Function

The Dirichlet function is defined such that it takes the value 1 at every rational point and the value 0 at every irrational point. This means that the function has different values based on the type of number (rational or irrational) that is input.

Step 2: Recall Continuity Definition

A function is continuous at a point if the limit of the function as it approaches the point from both directions equals the function's value at that point. Mathematically, this is expressed as \(\lim_{{x \to c}} f(x) = f(c)\).

Step 3: Analyze Continuity Points

For any given real number \(c\), any interval around \(c\) will have both rational and irrational numbers. Hence, for points around \(c\), \(f(x)\) will alternate between 0 and 1, based on whether \(x\) is rational or irrational. As a result, the limit does not exist at any point because it does not approach a single value.

Step 4: Conclude Discontinuity

Since the limit does not exist at any point, the function cannot be continuous at any real number. According to the definition of continuity, if a function is not continuous at any point \(c\), it is said to be discontinuous at all points.

Key Concepts

Discontinuity in Functions

The concept of discontinuity in functions relates to how a function does not have a consistent output as the input changes smoothly. For the Dirichlet function, this discontinuity is quite prominent. Here, the function takes on one value for rational numbers and a completely different value for irrational numbers. This means that as we move along the real number line trying to analyze the function, we jump between two distinct values: 1 and 0.

• If you're near a rational number onboard the real number line, the function value is 1.
• Close to an irrational number, the function value switches to 0.

These jumps or changes in output are clear indicators of discontinuity. The function does not smoothly transition from rational to irrational numbers and cannot achieve continuity anywhere along the real number line.

Rational and Irrational Numbers

To fully grasp the behavior of the Dirichlet function, we need to understand rational and irrational numbers. Rational numbers can be expressed as a fraction of two integers, such as \(\frac{3}{4}\), while irrational numbers cannot be so expressed and include numbers like \(\pi\) and \(\sqrt{2}\).
Since the function assigns different values to these two types of numbers:

• 1 for rational numbers
• 0 for irrational numbers

This bifurcation is the key reason behind the function's discontinuity everywhere. It inherently jumps between values when approaching any given real number. This unique property makes the Dirichlet function a textbook example in understanding discontinuity concerning rational and irrational numbers.

Continuity Rules

Continuity rules help determine if a function is smooth without any interruptions. A function is continuous at a point if the limit of the function as it approaches the point from both sides equals the function's value at that point. For the Dirichlet function, this rule is not satisfied anywhere.
As discussed earlier, due to the presence of both rational and irrational numbers around any given point, when moving closely toward any point on the real number line, the function does not converge to a single value. Therefore, the conditions for continuity are not met:

• No matter how small you make your interval around any point, you constantly flip between outputs of 0 and 1.
• This lack of a definitive single value means the function never aligns with the rules of continuity.

Thus, applying continuity rules, you can conclude that the Dirichlet function is discontinuous at every point on the real number line.