Let `f(x) ={:{(x, "for", 0 le x lt1),( 3-x,"for", 1 le x le2):}`
Then f(x) is

Options

A. continuous at x =1
B. Right continuous at x =1
C. Left continuous at x =1
D. Limit exists at x =1

Answer

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

\[ f(x) = \begin{cases} x & \text{for } 0 \leq x < 1 \\ 3 - x & \text{for } 1 \leq x \leq 2 \end{cases} \]

we need to check the continuity at the point \( x = 1 \).

### Step 1: Find the left-hand limit as \( x \) approaches 1.

The left-hand limit is given by:

\[ \lim_{x \to 1^-} f(x) \]

Since \( x \) is approaching 1 from the left, we use the first piece of the function \( f(x) = x \):

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^-} x = 1 \]

### Step 2: Find the right-hand limit as \( x \) approaches 1.

The right-hand limit is given by:

\[ \lim_{x \to 1^+} f(x) \]

Since \( x \) is approaching 1 from the right, we use the second piece of the function \( f(x) = 3 - x \):

\[ \lim_{x \to 1^+} f(x) = \lim_{x \to 1^+} (3 - x) = 3 - 1 = 2 \]

### Step 3: Find the value of the function at \( x = 1 \).

Now we evaluate the function at \( x = 1 \):

\[ f(1) = 3 - 1 = 2 \]

### Step 4: Check for continuity.

For the function to be continuous at \( x = 1 \), the following condition must hold:

\[ \lim_{x \to 1^-} f(x) = \lim_{x \to 1^+} f(x) = f(1) \]

From our calculations:

- Left-hand limit: \( \lim_{x \to 1^-} f(x) = 1 \)
- Right-hand limit: \( \lim_{x \to 1^+} f(x) = 2 \)
- Value of the function: \( f(1) = 2 \)

Since \( \lim_{x \to 1^-} f(x) \neq \lim_{x \to 1^+} f(x) \), the function is not continuous at \( x = 1 \).

### Conclusion:

- The function is **not continuous** at \( x = 1 \).
- It is **right continuous** at \( x = 1 \) since the right-hand limit equals the function value at that point.
- It is **not left continuous** at \( x = 1 \) since the left-hand limit does not equal the function value.
- The limit does not exist at \( x = 1 \) because the left-hand and right-hand limits are not equal.

### Final Answer:

The correct option is that \( f(x) \) is **right continuous at \( x = 1 \)**.

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