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?

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.