f(x)=(x-[x])sin(1/x), at x=0, f isA) Continuous B) Discontinuous C)not determined D) nonePlease answer as soon as possible.

To determine the continuity of the function \( f(x) = (x - [x]) \sin(1/x) \) at \( x = 0 \), we need to analyze its behavior as \( x \) approaches 0. Here, \( [x] \) denotes the greatest integer less than or equal to \( x \), also known as the floor function. Let's break this down step by step.

Understanding the Function Components

The function consists of two parts: \( (x - [x]) \) and \( \sin(1/x) \). The term \( (x - [x]) \) represents the fractional part of \( x \), which is always between 0 and 1 for any real number \( x \). As \( x \) approaches 0, \( [x] \) becomes 0 for values in the interval \( [0, 1) \), making \( (x - [x]) = x \) in this case.

Behavior of \( \sin(1/x) \)

The term \( \sin(1/x) \) oscillates between -1 and 1 as \( x \) approaches 0. This oscillation becomes more rapid as \( x \) gets closer to 0, leading to a situation where \( \sin(1/x) \) does not settle on any particular value. Thus, we need to consider the product of these two components.

Evaluating the Limit

To check for continuity at \( x = 0 \), we need to evaluate the limit:

• First, we find \( \lim_{x \to 0} f(x) = \lim_{x \to 0} (x - [x]) \sin(1/x) \).
• As \( x \) approaches 0 from the right, \( (x - [x]) \) approaches \( x \), which goes to 0.
• Even though \( \sin(1/x) \) oscillates, it is bounded between -1 and 1.

Thus, we can apply the squeeze theorem here. Since \( |(x - [x]) \sin(1/x)| \leq |x| \) and \( |x| \) approaches 0 as \( x \) approaches 0, we conclude:

\( \lim_{x \to 0} f(x) = 0 \).

Function Value at \( x = 0 \)

Next, we need to define \( f(0) \). Since the function is defined as \( f(x) = (x - [x]) \sin(1/x) \) for \( x \neq 0 \), we can set \( f(0) = 0 \) to ensure continuity. Therefore, we have:

• \( f(0) = 0 \)
• \( \lim_{x \to 0} f(x) = 0 \)

Final Assessment of Continuity

Since both the limit as \( x \) approaches 0 and the function value at \( x = 0 \) are equal, we can conclude that \( f(x) \) is continuous at \( x = 0 \).

In summary, the answer to whether \( f(x) \) is continuous at \( x = 0 \) is:

A) Continuous