Let `f(x) = (g(x))/(h(x))`, where g and h are continuous functions on the open interval (a, b). Which of the following statements is true for `a lt x lt b` ?

Options

A. (a)f is continuous at all x for which `x ne 0`
B. (b)f is continuous at all x for which g(x) = 0
C. (c)f is continuous at all x for which `g(x) ne 0`
D. (d)f is continuous at all x for which `h(x) ne 0`

Answer

To solve the problem, we need to analyze the function \( f(x) = \frac{g(x)}{h(x)} \), where \( g(x) \) and \( h(x) \) are continuous functions on the open interval \( (a, b) \). We want to determine which statement is true regarding the continuity of \( f(x) \) in the interval \( (a, b) \).

### Step-by-Step Solution:

1. **Understanding Continuity**:
- A function is continuous at a point if the limit of the function as it approaches that point is equal to the function's value at that point.
- For \( f(x) \) to be continuous, both \( g(x) \) and \( h(x) \) must be continuous, and \( h(x) \) must not be equal to zero.

2. **Condition for Division**:
- The function \( f(x) = \frac{g(x)}{h(x)} \) is defined and continuous wherever \( h(x) \neq 0 \).
- If \( h(x) = 0 \) at any point in the interval \( (a, b) \), then \( f(x) \) will be undefined at that point, leading to discontinuity.

3. **Conclusion**:
- Therefore, \( f(x) \) is continuous at all points \( x \) in the interval \( (a, b) \) where \( h(x) \neq 0 \).
- This leads us to the conclusion that the correct statement is: **"f is continuous at all x for which h(x) is not equal to 0."**

### Final Answer:
The correct option is **D: f is continuous at all x for which h(x) ≠ 0.**

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