The number of point (s) at which the function f(x)`=(1)/(x-[x])` is discontinuous is/are :

Options

A. 1
B. 2
C. 3
D. infinitely many

Answer

To determine the number of points at which the function \( f(x) = \frac{1}{x - [x]} \) is discontinuous, we can follow these steps:

### Step 1: Understand the function
The function \( f(x) = \frac{1}{x - [x]} \) involves the floor function \( [x] \), which represents the greatest integer less than or equal to \( x \). The term \( x - [x] \) gives us the fractional part of \( x \), denoted as \( \{x\} \).

### Step 2: Identify the domain
The function \( f(x) \) will be undefined when the denominator is zero. This occurs when:
\[ x - [x] = 0 \]
This happens when \( x \) is an integer because for any integer \( n \), \( [n] = n \) and thus \( n - [n] = 0 \).

### Step 3: Determine points of discontinuity
Since \( f(x) \) is undefined at all integer values of \( x \), we conclude that the function is discontinuous at these points. The set of integers is infinite.

### Step 4: Conclusion
Thus, the number of points at which the function \( f(x) \) is discontinuous is infinite.

### Final Answer
The number of points at which the function \( f(x) = \frac{1}{x - [x]} \) is discontinuous is infinitely many.

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