Statement-1 If `y=f(x)` is increasing in `[alpha,beta]`, then its range is `[f(alpha),f(beta)]`
Statement-2 Every increasing function need not to be continuous.

Options

A. Statement -1 is true, Statement -2 is true and Statement -2 is correct explanation for Statement -1
B. Statement -1 is true, Statement -2 is true and Statement -2 is not correct explanation for Statement -1
C. Statement -1 is true, Statement -2 is false
D. Statement -1 is false Statement -2 is true

Explanation

To analyze the given statements, let's break down the problem step by step.

### Step 1: Understanding Statement 1
Statement 1 claims that if \( y = f(x) \) is increasing on the interval \([ \alpha, \beta ]\), then its range is \([ f(\alpha), f(\beta) ]\).

**Analysis:**
- An increasing function means that if \( x_1 < x_2 \) in the interval \([ \alpha, \beta ]\), then \( f(x_1) < f(x_2) \).
- Therefore, the minimum value of \( f(x) \) on this interval will occur at \( x = \alpha \) and the maximum value will occur at \( x = \beta \), leading to the range being \([ f(\alpha), f(\beta) ]\).
- However, this assumes that \( f(x) \) is continuous on the interval \([ \alpha, \beta ]\). If there are discontinuities, the function could skip some values between \( f(\alpha) \) and \( f(\beta) \).

### Step 2: Considering Discontinuity
If \( f(x) \) is not continuous, it can have jumps or breaks. For example, if there is a point \( x_1 \) in the interval where \( f(x) \) takes a different value, the range may not cover all values from \( f(\alpha) \) to \( f(\beta) \).

**Example:**
- Let’s say \( f(x) \) is defined as follows:
- \( f(x) = x \) for \( x \in [\alpha, x_1) \)
- \( f(x) = c \) (a constant) at \( x_1 \)
- \( f(x) = x \) for \( x \in (x_1, \beta] \)
- Here, \( f(x) \) is increasing, but not continuous at \( x_1 \). The range could be limited to \([ f(\alpha), c ] \cup [ c, f(\beta) ]\) instead of \([ f(\alpha), f(\beta) ]\).

### Conclusion for Statement 1
Thus, Statement 1 is **false** because the range of an increasing function is not guaranteed to be \([ f(\alpha), f(\beta) ]\) if the function is not continuous.

### Step 3: Understanding Statement 2
Statement 2 states that every increasing function need not be continuous.

**Analysis:**
- This statement is true. As demonstrated in the previous example, an increasing function can have discontinuities and still maintain the property of being increasing.

### Conclusion for Statement 2
Thus, Statement 2 is **true**.

### Final Answer
- Statement 1 is false.
- Statement 2 is true.