Discuss the continuity of the function
f(x)`={(x", " x ge 0),(2", "x lt0):}` at x=0

Answer

To discuss the continuity of the function

\[ f(x) = \begin{cases} x & \text{if } x \geq 0 \\ 2 & \text{if } x < 0 \end{cases} \]

at \( x = 0 \), we need to check the following conditions:

1. **Left-hand limit** as \( x \) approaches 0.
2. **Right-hand limit** as \( x \) approaches 0.
3. The value of the function at \( x = 0 \).

### Step 1: Calculate the Left-hand Limit

The left-hand limit as \( x \) approaches 0 is given by:

\[ \lim_{x \to 0^-} f(x) \]

Since \( f(x) = 2 \) for \( x < 0 \), we have:

\[ \lim_{x \to 0^-} f(x) = 2 \]

### Step 2: Calculate the Right-hand Limit

The right-hand limit as \( x \) approaches 0 is given by:

\[ \lim_{x \to 0^+} f(x) \]

Since \( f(x) = x \) for \( x \geq 0 \), we have:

\[ \lim_{x \to 0^+} f(x) = \lim_{x \to 0^+} x = 0 \]

### Step 3: Calculate the Value of the Function at \( x = 0 \)

Now, we find the value of the function at \( x = 0 \):

\[ f(0) = 0 \]

### Step 4: Check Continuity Conditions

For the function \( f(x) \) to be continuous at \( x = 0 \), the following must hold:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0) \]

From our calculations:

- Left-hand limit: \( \lim_{x \to 0^-} f(x) = 2 \)
- Right-hand limit: \( \lim_{x \to 0^+} f(x) = 0 \)
- Value at \( x = 0 \): \( f(0) = 0 \)

Since the left-hand limit (2) does not equal the right-hand limit (0), we conclude that:

\[ \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x) \]

Thus, the function \( f(x) \) is discontinuous at \( x = 0 \).

### Final Conclusion

The function \( f(x) \) is discontinuous at \( x = 0 \).

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