If `f(x) =[|x|]` where `[.]` denotes the greatest integer function , then which of the following is not true ?

Options

A. f(x) is continuous `AA x in R`
B. f(x) is continuous from right and discontinuous from left `AA s in N`
C. f(x) is conttinuous from left and discontinuous from right `AA x in I`
D. f(x) is continuous at x=0

Answer

To solve the problem, we need to analyze the function \( f(x) = [|x|] \), where \([.]\) denotes the greatest integer function (also known as the floor function). We will evaluate the continuity of this function at various points and determine which statement is not true.

### Step 1: Analyze the function at integer points

1. **Understanding the function**: The function \( f(x) = [|x|] \) takes the absolute value of \( x \) and then applies the greatest integer function. This means that for any \( x \) in the interval \([n, n+1)\) where \( n \) is a non-negative integer, \( f(x) = n \).

2. **Check continuity at \( x = 1 \)**:
- **Right-hand limit**:
\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} [|x|] = [1] = 1 \]
- **Left-hand limit**:
\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} [|x|] = [1] = 1 \]
- **Value at point**:
\[ f(1) = [|1|] = [1] = 1 \]
- Since both limits equal the function value, \( f(x) \) is continuous at \( x = 1 \).

### Step 2: Check continuity at natural numbers

1. **For any natural number \( s \)**:
- **Right-hand limit**:
\[ \lim_{x \to s^+} f(x) = [|s|] = s \]
- **Left-hand limit**:
\[ \lim_{x \to s^-} f(x) = [|s| - \epsilon] = s - 1 \quad (\text{for small } \epsilon > 0) \]
- Since the right-hand limit equals \( s \) and the left-hand limit equals \( s - 1 \), the function is continuous from the right and discontinuous from the left at every natural number \( s \).

### Step 3: Check continuity at integer points

1. **For any integer \( n \)**:
- **Right-hand limit**:
\[ \lim_{x \to n^+} f(x) = [|n|] = n \]
- **Left-hand limit**:
\[ \lim_{x \to n^-} f(x) = [|n| - \epsilon] = n - 1 \]
- The function is continuous from the right and discontinuous from the left at every integer \( n \).

### Step 4: Check continuity at \( x = 0 \)

1. **At \( x = 0 \)**:
- **Right-hand limit**:
\[ \lim_{x \to 0^+} f(x) = [|0|] = [0] = 0 \]
- **Left-hand limit**:
\[ \lim_{x \to 0^-} f(x) = [|0|] = [0] = 0 \]
- **Value at point**:
\[ f(0) = [|0|] = [0] = 0 \]
- Since both limits equal the function value, \( f(x) \) is continuous at \( x = 0 \).

### Conclusion

Now let's summarize the statements:

1. **Statement 1**: \( f(x) \) is continuous for all \( x \in \mathbb{R} \) - **False** (discontinuous at integers).
2. **Statement 2**: \( f(x) \) is continuous from the right and discontinuous from the left for all \( s \in \mathbb{N} \) - **True**.
3. **Statement 3**: \( f(x) \) is continuous from the left and discontinuous from the right for all \( x \in \mathbb{Z} \) - **True**.
4. **Statement 4**: \( f(x) \) is continuous at \( x = 0 \) - **True**.

The statement that is **not true** is **Statement 1**.