1. The problem asks to analyze the continuity of the piecewise function: $$f(x) = \begin{cases} x^2, & x \leq 1 \\ 3 - x, & 1 < x \leq 2 \\ x, & x > 2 \end{cases}$$ 2. To check continuity at points where the definition changes ($x=1$ and $x=2$), we use the rule: A function $f$ is continuous at $x=c$ if: $$\lim_{x \to c^-} f(x) = f(c) = \lim_{x \to c^+} f(x)$$ 3. Check continuity at $x=1$: - Left limit: $\lim_{x \to 1^-} f(x) = 1^2 = 1$ - Function value: $f(1) = 1^2 = 1$ - Right limit: $\lim_{x \to 1^+} f(x) = 3 - 1 = 2$ Since left limit $\neq$ right limit, there is a jump discontinuity at $x=1$. 4. Check continuity at $x=2$: - Left limit: $\lim_{x \to 2^-} f(x) = 3 - 2 = 1$ - Function value: $f(2) = 3 - 2 = 1$ - Right limit: $\lim_{x \to 2^+} f(x) = 2$ Since left limit $\neq$ right limit, there is a jump discontinuity at $x=2$. 5. Therefore, your answer that the graph shows jump discontinuities at $x=1$ and $x=2$ is correct. 6. For the second problem, you describe three types of discontinuities: - Removable Discontinuity: parabola-like curve with a hole at $x=2$. - Jump Discontinuity: curve jumping from $y=0$ to $y=2$ at $x=0$. - Infinite Discontinuity: vertical asymptote at $x=0$ with curve approaching $\infty$ and $-\infty$. 7. Your descriptions correctly identify the three types of discontinuities. 8. Positioning the graphs as you described is a good way to visualize them. Final conclusion: Both your answers are correct.