1. **Problem Statement:** Given the graph of a function $f(x)$ with multiple segments and points, determine where the function is discontinuous, explain why, and classify the type of discontinuity. 2. **Recall the definition of continuity:** A function $f$ is continuous at a point $x=a$ if: - $f(a)$ is defined. - The limit $\lim_{x \to a} f(x)$ exists. - The limit equals the function value: $\lim_{x \to a} f(x) = f(a)$. If any of these fail, the function is discontinuous at $x=a$. 3. **Analyze the graph segments and points:** - At $x=-3$: There is an open circle at $(-3,-2)$ and a filled circle at $(-3,1)$. - The function value $f(-3)=1$ (filled circle). - The limit from the left approaches near $5$ (top-left curve), from the right approaches $-2$ (bottom-left line). - Left and right limits differ, so $\lim_{x \to -3} f(x)$ does not exist. - **Discontinuity:** Jump discontinuity at $x=-3$. - At $x=-2$: Open circle at $(-2,-1)$ and filled circle at $(-2,4)$. - $f(-2)=4$ (filled circle). - Left limit approaches $-1$, right limit approaches $4$. - Limits from left and right differ, so limit does not exist. - **Discontinuity:** Jump discontinuity at $x=-2$. - At $x=1$: Open circle at $(1,2)$, no filled circle shown. - $f(1)$ is not defined (no filled circle). - The limit from the left and right appears to approach some value (from graph, likely near 2). - Since $f(1)$ is undefined, function is discontinuous. - **Discontinuity:** Removable discontinuity at $x=1$. - At $x=2$: Filled circles at $(2,1)$ and $(2,6)$, indicating a jump. - $f(2)$ is defined at $1$ (filled circle on bottom-left segment). - The limit from the right is $6$ (top-right oscillating curve). - Left and right limits differ. - **Discontinuity:** Jump discontinuity at $x=2$. 4. **Summary:** - Discontinuities at $x=-3$, $x=-2$, $x=1$, and $x=2$. - Types: - $x=-3$: Jump discontinuity (limits from left and right differ). - $x=-2$: Jump discontinuity. - $x=1$: Removable discontinuity (function undefined but limit exists). - $x=2$: Jump discontinuity. 5. **Conclusion:** The function $f$ is discontinuous at $x=-3, -2, 1, 2$ due to jump discontinuities at $-3, -2, 2$ and a removable discontinuity at $1$.