Consider the piecewise defined function `f(x) = {{:(sqrt(-x),"if",x lt 0),(0,"if",0 le x le 4),(x - 4,"if",x gt 4):}`describe the continuity of this function.

Options

A. the function is unbounded and therefore cannot be continuous
B. the function is right continuous at x = 0
C. the function has a removable discontinuity at 0 and 4, but is continuous on the rest of the real line.
D. the function is continuous on the entire real line

Answer

To determine the continuity of the piecewise defined function

\[ f(x) = \begin{cases} \sqrt{-x} & \text{if } x < 0 \\ 0 & \text{if } 0 \leq x \leq 4 \\ x - 4 & \text{if } x > 4 \end{cases} \]

we need to check the continuity at the points where the definition of the function changes, specifically at \(x = 0\) and \(x = 4\).

### Step 1: Check continuity at \(x = 0\)

1. **Find the left-hand limit as \(x\) approaches 0:**
\[ \lim_{x \to 0^-} f(x) = \lim_{x \to 0^-} \sqrt{-x} = \sqrt{0} = 0 \]

2. **Find the right-hand limit as \(x\) approaches 0:**
\[ \lim_{x \to 0^+} f(x) = 0 \quad (\text{since } f(x) = 0 \text{ for } 0 \leq x \leq 4) \]

3. **Evaluate \(f(0)\):**
\[ f(0) = 0 \]

4. **Check if left-hand limit equals right-hand limit and \(f(0)\):**
\[ \lim_{x \to 0^-} f(x) = 0, \quad \lim_{x \to 0^+} f(x) = 0, \quad f(0) = 0 \]
Since all three values are equal, \(f(x)\) is continuous at \(x = 0\).

### Step 2: Check continuity at \(x = 4\)

1. **Find the left-hand limit as \(x\) approaches 4:**
\[ \lim_{x \to 4^-} f(x) = 0 \quad (\text{since } f(x) = 0 \text{ for } 0 \leq x \leq 4) \]

2. **Find the right-hand limit as \(x\) approaches 4:**
\[ \lim_{x \to 4^+} f(x) = \lim_{x \to 4^+} (x - 4) = 4 - 4 = 0 \]

3. **Evaluate \(f(4)\):**
\[ f(4) = 0 \]

4. **Check if left-hand limit equals right-hand limit and \(f(4)\):**
\[ \lim_{x \to 4^-} f(x) = 0, \quad \lim_{x \to 4^+} f(x) = 0, \quad f(4) = 0 \]
Since all three values are equal, \(f(x)\) is continuous at \(x = 4\).

### Step 3: Check continuity in other intervals

- For \(x < 0\), \(f(x) = \sqrt{-x}\) is continuous because it is defined as a square root function, which is continuous for non-negative inputs.
- For \(0 < x < 4\), \(f(x) = 0\) is a constant function, which is continuous.
- For \(x > 4\), \(f(x) = x - 4\) is a polynomial function, which is continuous everywhere.

### Conclusion

Since \(f(x)\) is continuous at \(x = 0\) and \(x = 4\), and continuous in the intervals \(x < 0\), \(0 < x < 4\), and \(x > 4\), we conclude that:

\[ \text{The function } f(x) \text{ is continuous for all } x \in \mathbb{R}. \]

---