Problem 56 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

Let \(f(x)\) be a continuous and \(g(x)\) be a discontinuous function. Prove that \(f(x)+g(x)\) is a discontinuous function.

Step-by-Step Solution

Answer

To prove that \(f(x) + g(x)\) is discontinuous if \(f(x)\) is continuous and \(g(x)\) is discontinuous, we first identify a discontinuity point \(x=c\) for g(x) and then evaluate the limits of \(f(x)\) and \(g(x)\) as \(x \to c\). Since \(f(x)\) is continuous, \(\lim_{x \to c} f(x) = f(c)\), and since \(g(x)\) is discontinuous, either \(\lim_{x \to c} g(x)\) does not exist or \(\lim_{x \to c} g(x) \neq g(c)\). Then, the limit of \(f(x) + g(x)\) as \(x \to c\) is \(f(c) + \lim_{x \to c} g(x)\), which, by applying the conditions of g(x) not being continuous, is not equal to \(f(c) + g(c)\), the value of \(f(x) + g(x)\) at x=c. Thus, \(f(x) + g(x)\) is discontinuous at the point x=c.

Step 1: Identify a discontinuity point in g(x)

Since g(x) is discontinuous, it means there exists a point, say x=c, at which either its limit does not exist or the limit is not equal to the value of g(x) at c.

Step 2: Evaluating the limit of f(x)

For f(x), as it is continuous at every point, we can write that as x approaches c, the limit of f(x) does exist and is equal to the function's value at c: \(\lim_{x \to c} f(x) = f(c)\)

Step 3: Evaluating the limit of g(x)

Given that g(x) is discontinuous at x =c, we know that one of these conditions must hold: 1. \(\lim_{x \to c} g(x)\) does not exist. 2. \(\lim_{x \to c} g(x) \neq g(c)\)

Step 4: Analyzing the limit of f(x) + g(x)

Now let's investigate the limit of f(x) + g(x) as x approaches c. We have: \(\lim_{x \to c} [f(x) + g(x)]\) Using the basic limit addition rule, this is equal to: \(\lim_{x \to c} f(x) + \lim_{x \to c} g(x)\) Since we know that \(\lim_{x \to c} f(x) = f(c)\), we can rewrite the above limit as: \(f(c) + \lim_{x \to c} g(x)\)

Step 5: Deducing the discontinuity of f(x) + g(x)

Since g(x) is discontinuous at x=c, one of the conditions from Step 3 will apply. So either the limit of g(x) as x approaches c does not exist, or is unequal to g(c). In either case, it implies that the limit of f(x) + g(x) as x approaches c cannot be equal to f(c) + g(c), which is the value of f(x) + g(x) at x=c. So, by definition, we can conclude that f(x) + g(x) is discontinuous at the point x=c.

Key Concepts

Continuous Functions

A continuous function is a function whose graph can be drawn without lifting the pencil off the paper. This means there are no breaks, jumps, or holes in the function. A formal way to express that a function is continuous at a point, say \(x = a\), involves limits:

• The limit of the function as \(x\) approaches \(a\) exists.
• This limit is equal to the function's value at that point, i.e., \(\lim_{x \to a} f(x) = f(a)\).

For example, the function \(f(x) = x^2\) is continuous everywhere because you can evaluate limits at every point, and they always equal the function's value.

Limits and Continuity

Limits and continuity are intimately connected. The concept of a limit focuses on what value a function approaches as the input approaches a particular point. For a function to be continuous at a point, its limit as it approaches that point must exist and equal the function's actual value at that point. Thus:

• Limits help us describe the behavior of functions near a point, while continuity ensures that behavior matches the specific value of the function at that point.
• A continuous function means the function’s value smoothly transitions without sudden changes or interruptions.

This relationship forms the core of calculus and allows us to analyze complex functions.

Sum of Functions

When you add two functions together, the resulting function can exhibit new properties. If you have two functions, say \(f(x)\) which is continuous, and \(g(x)\) which is discontinuous, and you consider their sum, \(f(x) + g(x)\), the discontinuities in \(g(x)\) can affect the continuity of the whole expression:

• The sum of a continuous and a discontinuous function may inherit the discontinuities from the discontinuous function.
• Unlike simple arithmetic, where you only add numbers, the behavior of functions when added involves their limits and continuity properties.

Thus, the combination of properties from each function results in a new function often dominated by the discontinuous traits.

Discontinuity Point

A discontinuity point in a function is where the function is not continuous. This could mean:

• The limit from the left and the limit from the right do not agree.
• The limit does not exist at that point.
• The limit exists but is not equal to the function's value at that point.

These points represent breaks or jumps in the graph of a function. In analyzing functions, pinpointing where and why a function is discontinuous (as with \(g(x)\) at \(x = c\)) helps us understand its behavior and how it may affect operations involving the function, such as its sum with a continuous function.