If `f(x)={:{(3", if " 0 le x le 1),(4", if " 1 lt x lt 3),(5", if " 3 le x le 10):}`, then

Options

A. f is continuous on [0, 10] except at `x=1, 3`
B. f is continuous on [0, 10] except at `x=1`
C. f is continuous on [0, 10] except at `x=3`
D. f is continuous on [0, 10]

Answer

To determine the continuity of the piecewise function

\[ f(x) = \begin{cases} 3 & \text{if } 0 \leq x \leq 1 \\ 4 & \text{if } 1 < x < 3 \\ 5 & \text{if } 3 \leq x \leq 10 \end{cases} \]

we need to check the continuity at the points where the function changes its definition, specifically at \(x = 1\) and \(x = 3\).

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

1. **Find \(f(1)\)**:
Since \(1\) is included in the first case, we have:
\[ f(1) = 3 \]

2. **Find the left-hand limit as \(x\) approaches \(1\)**:
\[ \lim_{x \to 1^-} f(x) = 3 \quad (\text{since } f(x) = 3 \text{ for } 0 \leq x \leq 1) \]

3. **Find the right-hand limit as \(x\) approaches \(1\)**:
\[ \lim_{x \to 1^+} f(x) = 4 \quad (\text{since } f(x) = 4 \text{ for } 1 < x < 3) \]

4. **Compare limits and function value**:
\[ \lim_{x \to 1^-} f(x) = 3 \quad \text{and} \quad \lim_{x \to 1^+} f(x) = 4 \]
Since \(3 \neq 4\), \(f(x)\) is not continuous at \(x = 1\).

### Step 2: Check continuity at \(x = 3\)

1. **Find \(f(3)\)**:
Since \(3\) is included in the third case, we have:
\[ f(3) = 5 \]

2. **Find the left-hand limit as \(x\) approaches \(3\)**:
\[ \lim_{x \to 3^-} f(x) = 4 \quad (\text{since } f(x) = 4 \text{ for } 1 < x < 3) \]

3. **Find the right-hand limit as \(x\) approaches \(3\)**:
\[ \lim_{x \to 3^+} f(x) = 5 \quad (\text{since } f(x) = 5 \text{ for } 3 \leq x \leq 10) \]

4. **Compare limits and function value**:
\[ \lim_{x \to 3^-} f(x) = 4 \quad \text{and} \quad \lim_{x \to 3^+} f(x) = 5 \]
Since \(4 \neq 5\), \(f(x)\) is not continuous at \(x = 3\).

### Conclusion

The function \(f(x)\) is continuous on the intervals \( (0, 1) \) and \( (1, 3) \) and \( (3, 10) \), but it is not continuous at \(x = 1\) and \(x = 3\). Therefore, we conclude that:

- **f is continuous on \( [0, 10] \) except at \(x = 1\) and \(x = 3\)**.