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

The Correct Option is B

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