Discuss the continuity of the function
f(x)`={(3-x", "x le 0),(x", "x gt 0):}` at x=0.

Answer

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

\[ f(x) = \begin{cases} 3 - x & \text{if } x \leq 0 \\ x & \text{if } x > 0 \end{cases} \]

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

1. \( f(0) \) must be defined.
2. The left-hand limit \( \lim_{x \to 0^-} f(x) \) must exist.
3. The right-hand limit \( \lim_{x \to 0^+} f(x) \) must exist.
4. The left-hand limit and right-hand limit must be equal to \( f(0) \).

### Step 1: Find \( f(0) \)

Since \( x = 0 \) falls under the first case of the piecewise function:

\[ f(0) = 3 - 0 = 3 \]

### Step 2: Find the left-hand limit \( \lim_{x \to 0^-} f(x) \)

For \( x < 0 \), we use the first case of the function:

\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} (3 - x) = 3 - 0 = 3 \]

### Step 3: Find the right-hand limit \( \lim_{x \to 0^+} f(x) \)

For \( x > 0 \), we use the second case of the function:

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

### Step 4: Compare the limits and \( f(0) \)

Now we compare the values we have found:

- \( f(0) = 3 \)
- \( \lim_{x \to 0^-} f(x) = 3 \)
- \( \lim_{x \to 0^+} f(x) = 0 \)

Since \( \lim_{x \to 0^-} f(x) = 3 \) and \( \lim_{x \to 0^+} f(x) = 0 \), we see that:

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

Thus, the left-hand limit does not equal the right-hand limit. Therefore, the function \( f(x) \) is not continuous at \( x = 0 \).

### Conclusion

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

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