1. **State the problem:** We want to find the points $(x,y)$ where the function $$f(x,y) = \frac{x^2 y + y^3}{x^2 + y^2 + 1}$$ is continuous. 2. **Analyze the function:** The function is a rational function where the numerator is $x^2 y + y^3$ and the denominator is $x^2 + y^2 + 1$. 3. **Check the denominator:** The denominator is $x^2 + y^2 + 1$. Since $x^2 \geq 0$ and $y^2 \geq 0$ for all real $x,y$, and $1 > 0$, the denominator is always positive: $$x^2 + y^2 + 1 \geq 1 > 0$$ 4. **Conclusion on continuity:** Because the denominator never equals zero, the function has no points of discontinuity due to division by zero. 5. **Continuity of numerator and denominator:** Both numerator and denominator are polynomials in $x$ and $y$, which are continuous everywhere. 6. **Final answer:** Since the denominator is never zero and numerator and denominator are continuous everywhere, the function $f(x,y)$ is continuous for all real values of $x$ and $y$. **Answer:** The function $f(x,y)$ is continuous for all points $(x,y) \in \mathbb{R}^2$.