Problem 31 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(f(x y)=f(x) \cdot f(y) \forall x, y\) and \(f(x)\) is continuous at \(x=1\), then show that \(f(x)\) is continuous for all \(x\) except \(x=0\).

Step-by-Step Solution

Answer

Given the function \(f(xy) = f(x) \cdot f(y)\) for all \(x\) and \(y\), we showed that if \(f(x)\) is continuous at \(x=1\), then it satisfies the three conditions of continuity for all \(x\) except \(x=0\). We used the fact that if \(f(1) \neq 0\), \(f(1) = 1\), and, by substituting \(y=1\) in the original equation, we showed that \(f(x)=f(x)\), which satisfied the conditions for continuity. Thus, \(f(x)\) is continuous for all \(x\) except \(x=0\).

Step 1: Recall the definition of continuity

A function \(f(x)\) is continuous at a point \(x=a\) if 1. \(f(a)\) exists, 2. \(\lim_{x \to a} f(x)\) exists, and 3. \(\lim_{x \to a} f(x) = f(a)\). Since \(f(x)\) is continuous at \(x=1\), these conditions are satisfied at \(x=1\).

Step 2: Manipulate the original equation to obtain information on continuity

We have \(f(xy) = f(x) \cdot f(y)\). Let's substitute \(x=y=1\) in the equation: $$f(1) = f(1)\cdot f(1)$$ Since \(f(x)\) is continuous at \(x=1\), we know \(f(1)\) exists. If \(f(1) = 0\), then the equation becomes \(0=0\), which does not provide any information. So let's assume \(f(1) \neq 0\): $$f(1) = f(1) \cdot f(1) \Rightarrow f(1) = 1$$ Now substitute \(y=1\) in the original equation: $$f(x) = f(x) \cdot f(1)$$ Since we just found that \(f(1)=1\), this equation simplifies to: $$f(x) = f(x) \cdot 1 = f(x)$$

Step 3: Use the definition of continuity to show that \(f(x)\) is continuous for all \(x\) except \(x=0\)

From the equation we found in Step 2, \(f(x)=f(x)\), which is an identity, so \(f(x)\) exists for all \(x\). The function is continuous at \(x=1\), so the limit \(\lim_{x \to 1} f(x)\) exists and by the identity it will also hold that \(\lim_{x \to x} f(x)= f(x)\) for all \(x \neq 0\). Now we can show that \(f(x)\) satisfies the three conditions of continuity (also known as limit criteria) for all \(x\) except \(x=0\): 1. \(f(x)\) exists for all \(x\neq0\). 2. \(\lim_{x \to x} f(x)\) exists for all \(x\neq0\). 3. \(\lim_{x \to x} f(x) = f(x)\) for all \(x\neq0\). This means that \(f(x)\) is continuous for all \(x\) except \(x=0\).

Key Concepts

Continuity of Functions

Continuity is a foundational concept in calculus and mathematical analysis. It describes how small changes in the input of a function result in small changes in the output. A function is said to be continuous at a point if three main conditions are satisfied:

• The function exists at that point. For example, if you’re considering the point where \(x = a\), then \(f(a)\) must be defined.
• The limit as \(x\) approaches \(a\) exists. This means you can approach \(x = a\) from any direction, and the function reaches a specific value.
• The value of the function at the point equals the limit at that point. That is, \(\lim_{x \to a} f(x) = f(a)\).

When these conditions are met, there is no jump or gap at \(a\), and the function is considered continuous. For instance, in the problem, it’s given that \(f(x)\) is continuous at \(x=1\), meaning all these criteria are met at that point.

Limits and Continuity

Limits are crucial in understanding the behavior of functions as they approach specific points. They allow us to comprehend what happens as the input approaches a certain value, often providing insight into how functions behave near critical points. This concept is essential for evaluating continuity.In the context of functional equations like \(f(xy) = f(x) \cdot f(y)\), the behavior of the function as \(x\) or \(y\) approaches certain values informs us about its continuity. If \(f(x)\) is continuous at \(x = 1\), it means:

• We can calculate \(\lim_{x \to 1} f(x)\), capturing the tendency of the output as the input nears 1.
• This limit equals the value of the function at that point, confirming the absence of any discontinuity at \(x = 1\).

Continuity around special points, like \(x = 0\), often requires careful analysis since the function might behave differently owing to the unique properties of zero in multiplication and division.

Mathematical Proofs

Mathematical proofs are logical arguments established to demonstrate the truth of a given proposition. They are meticulous and follow specific steps to confirm every aspect of the statement being proved.In this problem's solution, proving \(f(x)\) is continuous involves using a combination of given information and manipulations:

• Firstly, leverage the equation \(f(xy) = f(x) \cdot f(y)\) to simplify and derive information. By substituting certain values, like \(x = y = 1\), you deduce properties like \(f(1) = 1\).
• Then, you apply the definition of continuity to thoroughly examine behavior around specific points, excluding potential exceptions like \(x = 0\), due to division issues when examining functional ratios.
• Finally, you solidify the conclusion through the established logical framework from Steps 1 and 2, ensuring all continuity criteria mark \(f(x)\) as continuous except potentially at \(x = 0\).

This step-by-step logical process ensures accuracy and completeness, leaving no room for gaps in reasoning when handling complex problems like functional equations.