The functions `g(x)=x-[x]` is discontinuous at x =

Options

A. `-2.1`
B. `1/2`
C. 0
D. 1.2

Answer

To determine the points of discontinuity of the function \( g(x) = x - [x] \), where \([x]\) is the greatest integer function, we will analyze the continuity of \( g(x) \) at integer points.

### Step 1: Identify the function
The function is given as:
\[ g(x) = x - [x] \]
This function represents the fractional part of \( x \), which is continuous everywhere except at integer points.

### Step 2: Check for continuity at integer points
To check for continuity at an integer point \( x = n \) (where \( n \) is an integer), we need to evaluate the left-hand limit (LHL), right-hand limit (RHL), and the function value at that point.

### Step 3: Calculate \( g(n) \)
For any integer \( n \):
\[ g(n) = n - [n] = n - n = 0 \]

### Step 4: Calculate the left-hand limit as \( x \) approaches \( n \) from the left
\[ \text{LHL at } n = \lim_{x \to n^-} g(x) = \lim_{x \to n^-} (x - [x]) \]
As \( x \) approaches \( n \) from the left (e.g., \( n - 0.1 \)):
\[ [x] = n - 1 \quad \text{(since \( x < n \))} \]
Thus,
\[ g(n^-) = (n - 0.1) - (n - 1) = 0.9 \]

### Step 5: Calculate the right-hand limit as \( x \) approaches \( n \) from the right
\[ \text{RHL at } n = \lim_{x \to n^+} g(x) = \lim_{x \to n^+} (x - [x]) \]
As \( x \) approaches \( n \) from the right (e.g., \( n + 0.1 \)):
\[ [x] = n \quad \text{(since \( x \geq n \))} \]
Thus,
\[ g(n^+) = (n + 0.1) - n = 0.1 \]

### Step 6: Compare the limits and function value
Now we compare:
- \( g(n) = 0 \)
- LHL at \( n = 0.9 \)
- RHL at \( n = 0.1 \)

Since:
\[ \text{LHL} \neq \text{RHL} \quad (0.9 \neq 0.1) \]
and
\[ \text{LHL} \neq g(n) \quad (0.9 \neq 0) \]

### Conclusion
The function \( g(x) \) is discontinuous at every integer point \( n \).

### Final Answer
The function \( g(x) = x - [x] \) is discontinuous at \( x = n \) where \( n \) is any integer.

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