Which one of the following is true ?
For the real function :
`f(x)={{:(x+2" if "xle1),(x-2" if "xgt1):}`,

Options

A. f is continuous at all real numbers `xgt1andxlt1`
B. f is continuous at all real numbers `xge1`
C. f is continuous at all real numbers `xle1`
D. f is continuous at x = 1.

Answer

To determine the continuity of the function \( f(x) \) defined as:

\[ f(x) = \begin{cases} x + 2 & \text{if } x \leq 1 \\ x - 2 & \text{if } x > 1 \end{cases} \]

we will analyze the function in two parts and check its continuity at the point where the definition changes, which is at \( x = 1 \).

### Step 1: Check continuity for \( x < 1 \)

For \( x < 1 \), the function is defined as \( f(x) = x + 2 \).

- This is a linear function and is continuous everywhere in its domain. Therefore, \( f(x) \) is continuous for all \( x < 1 \).

**Hint:** Linear functions are continuous everywhere.

### Step 2: Check continuity for \( x > 1 \)

For \( x > 1 \), the function is defined as \( f(x) = x - 2 \).

- Again, this is a linear function and is continuous everywhere in its domain. Therefore, \( f(x) \) is continuous for all \( x > 1 \).

**Hint:** Linear functions are continuous everywhere.

### Step 3: Check continuity at \( x = 1 \)

To check continuity at \( x = 1 \), we need to evaluate:

1. \( f(1) \)
2. The left-hand limit as \( x \) approaches 1
3. The right-hand limit as \( x \) approaches 1

**Calculating \( f(1) \):**
\[ f(1) = 1 + 2 = 3 \]

**Calculating the left-hand limit:**
\[ \lim_{h \to 0^-} f(1 + h) = \lim_{h \to 0^-} f(1) = f(1) = 3 \]

**Calculating the right-hand limit:**
\[ \lim_{h \to 0^+} f(1 + h) = \lim_{h \to 0^+} f(1 + h) = \lim_{h \to 0^+} (1 + h - 2) = 1 - 2 = -1 \]

### Step 4: Compare the limits and the function value

- Left-hand limit: \( 3 \)
- Right-hand limit: \( -1 \)
- Value at \( x = 1 \): \( 3 \)

Since the left-hand limit \( (3) \) is not equal to the right-hand limit \( (-1) \), the function is discontinuous at \( x = 1 \).

### Conclusion

The function \( f(x) \) is continuous for all \( x \) except at \( x = 1 \). Therefore, the correct statement is that \( f \) is continuous at all real numbers except at \( x = 1 \).

**Final Answer:** The function is discontinuous at \( x = 1 \).