Problem 243 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Test for continuity and differentiability of the function \(\&\) find \(f^{\prime}(x)\) \(f(x)=x, \quad x\) is rational \(=-x, \quad x\) is irrational.

Step-by-Step Solution

Answer

The function is not continuous and not differentiable for all real numbers \(x\). The derivative of the function does not exist for any real \(x\).

Step 1: Analyze the Continuity of the Function

The function is given by \(f(x) = x\) when \(x\) is rational, and \(f(x) = -x\) when \(x\) is irrational. These are both linear functions and are individually continuous everywhere. Therefore, the question of continuity depends on whether the values of the two parts of the function are the same at the transition points between rational and irrational numbers. As there are no clear transitions from a rational to an irrational number, the function is discontinuous for all real numbers \(x\).

Step 2: Analyze the Differentiability of the Function

A function is differentiable at a certain point if it is continuous at that point and if the derivative exists at that point. Since the function is discontinuous for every real number \(x\), it is not differentiable for every real number \(x\).

Step 3: Find the Derivative of the Function

The given function is actually two separate functions, one for rational \(x\) and one for irrational \(x\). For \(f(x) = x\) when \(x\) is rational, the derivative \(f^{\prime}(x)\) is 1, and for \(f(x) = -x\) when \(x\) is irrational, the derivative \(f^{\prime}(x)\) is -1. However, since we have already established that the function is not differentiable for any real number, the overall derivative \(f^{\prime}(x)\) does not exist for any real \(x\).

Key Concepts

Rational and Irrational Numbers

When exploring functions, rational and irrational numbers play crucial roles. Rational numbers can be expressed as a fraction, where both the numerator and the denominator are integers, and the denominator is not zero. Examples include \( \frac{1}{2} \), \( 0.7 \), and \(-3\).

• They include integers, fractions, and terminating or repeating decimals.

On the other hand, irrational numbers cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions. Numbers like \(\pi\), \(e\), and \(\sqrt{2}\) are irrational.

• The division of these numbers into separate expressions can cause unique behaviors in functions, such as potential discontinuities.

Discontinuous Function

In mathematical analysis, continuity implies the absence of any breaks, jumps, or holes in the graph of the function. For a function to be continuous at a point, the left-hand limit, the right-hand limit, and the actual function value at that point must be equal.
In this particular problem, the function is defined as \( f(x) = x \) for rational \(x\) and \( f(x) = -x \) for irrational \(x\). This creates a problem because any real number \(x\) can be approached by both rational and irrational numbers.

• The values differ based on whether \(x\) is rational or irrational, leading to discontinuity.
• There isn't a smooth transition or an interval where these values coincide.

Without continuity, a function cannot have a derivative. This discontinuity means the function behaves unexpectedly between the two cases and lacks the required smoothness for differentiability.

Derivative Calculation

When finding the derivative of a function at a point, it typically requires continuity at that point. For separate linear functions, derivatives do exist:

• The derivative of \(f(x) = x\) for rational \(x\) is \(f'(x) = 1\).
• The derivative of \(f(x) = -x\) for irrational \(x\) is \(f'(x) = -1\).

However, since the function \(f(x)\) is not continuous for any real number \(x\), a smooth derivative across the domain cannot be established.

• Continuity is a prerequisite for differentiability; without it, the derivative is undefined for the overall function across any real point.
• The meaning of having \(f'(x) = 1\) and \(f'(x) = -1\) is for illustrative purposes; they don't apply to the whole function.

Therefore, this function shows that differentiability issues can arise in functions with separate definitions, which differ based on the type of number, impacting how derivatives can be calculated.