Question

Let $f_i: \mathbb{R} \to \mathbb{R}$ for $i = 1, 2$ be defined as follows:
$f_1(x) = \begin{cases} \sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right), & \text{if } x \neq 0 \\\0, & \text{if } x = 0 \end{cases}$
and
$f_2(x) = \begin{cases} x\left(\sin\left(\frac{1}{x}\right) + \cos\left(\frac{1}{x}\right)\right), & \text{if } x \neq 0 \\\0, & \text{if } x = 0 \end{cases}$
Then, determine the continuity of these functions at $x = 0$:

• A. $f_1$ is continuous at $0$ but $f_2$ is not continuous at $0$
• B. $f_1$ is not continuous at $0$ but $f_2$ is continuous at $0$
• C. Both $f_1$ and $f_2$ are continuous at $0$
• D. Neither $f_1$ nor $f_2$ is continuous at $0$

Answer 1

Answer: (B)

$f_1$ is not continuous at $0$ but $f_2$ is continuous at $0$

Answer 2

The correct option is (B): $f_1$ is not continuous at $0$ but $f_2$ is continuous at $0$