discontinuity at f(x) = 3csc(x^2) on the interval [0,pi]

To analyze the discontinuity of the function \( f(x) = 3 \csc(x^2) \) on the interval \( [0, \pi] \), we need to recall the definition of the cosecant function and where it is undefined. The cosecant function is defined as \( \csc(x) = \frac{1}{\sin(x)} \), and it is undefined where \( \sin(x) = 0 \).

However, in this case, we are dealing with \( \csc(x^2) \), meaning we need to examine where \( \sin(x^2) = 0 \) on the interval \( [0, \pi] \).

### Step 1: Analyze \( \sin(x^2) = 0 \)
The function \( \sin(x^2) = 0 \) when \( x^2 = n\pi \), where \( n \) is an integer. This implies that \( x = \sqrt{n\pi} \).

We are particularly interested in values of \( x \) within the interval \( [0, \pi] \), so we look for values of \( n \) such that \( \sqrt{n\pi} \) lies in this interval. Let's check:

- For \( n = 0 \): \( \sqrt{0\pi} = 0 \).
- For \( n = 1 \): \( \sqrt{1\pi} = \sqrt{\pi} \approx 1.77 \), which is within \( [0, \pi] \).

Thus, \( \sin(x^2) = 0 \) when \( x = 0 \) and \( x = \sqrt{\pi} \), which is approximately \( 1.77 \).

### Step 2: Identify the discontinuities
The function \( f(x) = 3 \csc(x^2) = \frac{3}{\sin(x^2)} \) will have discontinuities wherever \( \sin(x^2) = 0 \), which occurs at \( x = 0 \) and \( x = \sqrt{\pi} \). These are the points where the function is undefined.

- **At \( x = 0 \)**: The function has a **discontinuity**, as \( \csc(0^2) \) is undefined (division by zero).
- **At \( x = \sqrt{\pi} \)**: The function also has a **discontinuity**, as \( \sin(\sqrt{\pi}^2) = \sin(\pi) = 0 \), and thus \( \csc(\pi) \) is undefined.

### Conclusion:
The function \( f(x) = 3 \csc(x^2) \) has discontinuities at \( x = 0 \) and \( x = \sqrt{\pi} \) on the interval \( [0, \pi] \).

Would you like more details or have any questions about this analysis?

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Here are 5 related questions to expand on this topic:
1. How do you determine if a discontinuity is removable or not?
2. What is the behavior of \( \csc(x) \) near its discontinuities?
3. How do you find asymptotes for functions involving \( \csc(x) \)?
4. What is the general form of discontinuities for trigonometric functions?
5. How does the domain of a function change when introducing discontinuities?

**Tip**: When analyzing discontinuities, always check the behavior of the function near the problematic points by using limits.

Analyze the discontinuities in the function f(x) = 3csc(x^2) on the interval [0, π]. Explore key trigonometric concepts, such as the behavior of the cosecant function and where it becomes undefined. This advanced problem is ideal for college calculus or high school students learning about trigonometric discontinuities.

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