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To check whether the function \( f(x) \) is continuous on \( \mathbb{R} \), we need to verify the following:

1. **Continuity at all points where \( x \geq 0 \):** This would involve analyzing the function \( \frac{3\sqrt{x}}{x^2 - x + 1} \) for \( x \geq 0 \).
   
2. **Continuity at all points where \( x < 0 \):** We need to check the continuity of \( x^2 \sin(x - 2) \) for \( x < 0 \).

3. **Continuity at the point \( x = 0 \),** where the two parts of the piecewise function meet.

### Step-by-step analysis:

#### 1. Continuity for \( x \geq 0 \):
The first part of the function is \( f(x) = \frac{3\sqrt{x}}{x^2 - x + 1} \) for \( x \geq 0 \).

- For \( x \geq 0 \), both the numerator \( 3\sqrt{x} \) and the denominator \( x^2 - x + 1 \) are continuous functions.
- The denominator \( x^2 - x + 1 \) does not vanish for any real \( x \), ensuring there are no singularities in this part of the function.

Thus, \( f(x) \) is continuous for \( x \geq 0 \).

#### 2. Continuity for \( x < 0 \):
The second part of the function is \( f(x) = x^2 \sin(x - 2) \) for \( x < 0 \).

- This is the product of a continuous function \( x^2 \) and a continuous trigonometric function \( \sin(x - 2) \).
- Therefore, this part is continuous for all \( x < 0 \).

#### 3. Continuity at \( x = 0 \):
To check the continuity at \( x = 0 \), we need to confirm that:
\[
\lim_{x \to 0^-} f(x) = \lim_{x \to 0^+} f(x) = f(0)
\]

##### For \( x \to 0^+ \):
\[
\lim_{x \to 0^+} \frac{3\sqrt{x}}{x^2 - x + 1} = \frac{3\sqrt{0}}{0^2 - 0 + 1} = 0
\]

##### For \( x \to 0^- \):
\[
\lim_{x \to 0^-} x^2 \sin(x - 2) = 0^2 \cdot \sin(0 - 2) = 0
\]

Both limits are equal to 0, and the value of the function at \( x = 0 \) (from either side) is also 0. Thus, \( f(0) = 0 \).

### Conclusion:
Since the function is continuous on both intervals and the left-hand and right-hand limits at \( x = 0 \) are equal, \( f(x) \) is continuous on \( \mathbb{R} \).

Would you like more details or have any questions?

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Here are 5 related questions you might find helpful:
1. How do you check differentiability for piecewise functions like this one?
2. What are some examples of functions that are continuous but not differentiable?
3. How do we ensure a function is smooth (continuously differentiable)?
4. Can this function be integrated over all real numbers, and how do we evaluate that?
5. How do you handle discontinuities or singularities in a function?

**Tip:** When evaluating continuity at a point, always compare the left-hand and right-hand limits, as well as the function's value at that point.

Explore the continuity of a piecewise function involving square roots and trigonometric expressions. This solution walks through verifying continuity for x ≥ 0, x < 0, and at the critical point x = 0 using limits. Suitable for undergraduate mathematics students.

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