Question

If \(f(x)\) has isolated point discontinuity at \(x=a\) such that \(|f(x)|\) is continuous at \(x=\) a then (A) \(|\mathrm{f}(\mathrm{x})|\) must be differentiable at \(\mathrm{x}=\mathrm{a}\) (B) \(\lim _{x \rightarrow a} f(x)\) does not exist (C) \(\lim _{x \rightarrow a} f(x)+f(a)=0\) (D) \(\mathrm{f}(\mathrm{a})=0\)

Short answer

(A) |f(x)| must be differentiable at x=a. (B) \(\lim_{x \rightarrow a} f(x)\) does not exist. (C) \(\lim_{x \rightarrow a} f(x) + f(a) = 0\). (D) f(a)=0. Answer: None of the statements (A), (B), (C), and (D) are necessarily correct based on the given information.

Step-by-step solution

Analyze continuity and discontinuity properties

If a function is continuous at a point x=a, it means the function satisfies the following three conditions: 1. f(a) is defined 2. \(\lim_{x \rightarrow a} f(x)\) exists 3. f(a) = \(\lim_{x \rightarrow a} f(x)\) If any of these conditions are not met, the function is said to have a discontinuity at x=a. In this exercise's case, f(x) has an isolated point discontinuity at x=a.

Analyze the properties of the given function

Given that f(x) has isolated point discontinuity at x=a, it means f(a) is defined but the limit of f(x) as x approaches a may or may not exist. Since we are given that |f(x)| is continuous at x=a, it follows that: 1. |f(a)| is defined 2. \(\lim_{x \rightarrow a} |f(x)|\) exists 3. |f(a)| = \(\lim_{x \rightarrow a} |f(x)|\)

Analyze statement (A)

The question asks if \(|\mathrm{f}(\mathrm{x})|\) must be differentiable at \(x=a\). Knowing that |f(x)| is continuous at x=a, we can't conclude that it must be differentiable, as continuity does not necessarily imply differentiability. Therefore, statement (A) is not necessarily correct.

Analyze statement (B)

Statement (B) states that \(\lim _{x \rightarrow a} f(x)\) does not exist. However, based on the given information, we can't definitely conclude whether this limit exists or not. The isolated point discontinuity only implies that f(a) is not equal to the limit, not that the limit doesn't exist. Therefore, statement (B) is not necessarily correct.

Analyze statement (C)

For statement (C), we have to check if \(\lim _{x \rightarrow a} f(x)+f(a)=0\). Now, if the limit of f(x) as x approaches a exists, then \(\lim _{x \rightarrow a} f(x) = L\), for some L. It follows that \(\lim _{x \rightarrow a} f(x) + f(a) = L + f(a)\). However, since f(x) has an isolated point discontinuity at x=a, L + f(a) can't be equal to f(a), and the statement does not hold in this case. If the limit doesn't exist, then statement (C) also doesn't hold. Therefore, statement (C) is not necessarily correct.

Analyze statement (D)

Statement (D) asks if f(a)=0. Based on the given information, we know that |f(a)| = \(\lim_{x \rightarrow a} |f(x)|\). But it's not enough to conclude that f(a) must be zero. It could be that f(a) is any other value, and |f(a)| is still equal to the limit. Therefore, statement (D) is not necessarily correct. In conclusion, none of the statements (A), (B), (C), and (D) are necessarily correct based on the given information.