1. **Problem Statement:** Identify points of discontinuity in the given graphs (a), (b), (c) and analyze continuity/discontinuity of given piecewise and domain-restricted functions. 2. **Discontinuity in Graphs:** - Graph (a): There is an open circle near $x=0$ indicating a jump or removable discontinuity. - Graph (b): Open circle on the right side of the parabola indicates discontinuity at that point. - Graph (c): Open circle near the middle shows discontinuity. 3. **Continuity and Discontinuity of Functions:** - Functions defined on natural numbers $\mathbb{N}$, whole numbers $\mathbb{W}$, and integers $\mathbb{Z}$ are discrete, so continuity is considered only at those points. 4. **Function Analysis:** (a) $y = x + 1$ for $x \in \mathbb{N}$: Continuous on $\mathbb{N}$ since it is a linear function defined on discrete points. (b) $f(x) = 3x - 2$ for $x \in \mathbb{W}$: Continuous on $\mathbb{W}$ for same reasons. (c) $y = 7 - 2x$ for $x \in \mathbb{Z}$: Linear and continuous on integers. (d) $y = x^2$ for $x \in \mathbb{Z}$: Polynomial continuous on integers. (e) $f(x) = 2x^2 + 3$ for $x \in \mathbb{Z}$: Polynomial continuous on integers. (f) $g(x) = \frac{x + 2}{3}$ for $x \in \mathbb{Z}$: Linear continuous on integers. (g) Piecewise function: $$y = \begin{cases} x + 1 & x \geq 3 \\ 3x - 5 & x < 3 \end{cases}$$ Check continuity at $x=3$: $$\lim_{x \to 3^-} 3x - 5 = 3(3) - 5 = 4$$ $$\lim_{x \to 3^+} x + 1 = 3 + 1 = 4$$ Function value at $x=3$ is $3 + 1 = 4$, so continuous. (h) Piecewise function: $$y = \begin{cases} 3x + 1 & x \geq 1 \\ x + 2 & x < 1 \end{cases}$$ Check at $x=1$: $$\lim_{x \to 1^-} x + 2 = 3$$ $$\lim_{x \to 1^+} 3x + 1 = 4$$ Limits differ, so discontinuous at $x=1$. (i) Piecewise function: $$y = \begin{cases} \sin x & x > 0 \\ x^3 & x < 0 \end{cases}$$ No definition at $x=0$, so discontinuous there. (j) $g(x) = \frac{x^2 - 9}{x - 3}$ for $x \in \mathbb{Z}$: Factor numerator: $$x^2 - 9 = (x - 3)(x + 3)$$ Simplify: $$g(x) = x + 3, x \neq 3$$ At $x=3$, denominator zero, so discontinuous. (k) $g(x) = \frac{x^2 - 9}{x - 1}$ for $x \in \mathbb{Z}$: Denominator zero at $x=1$, numerator $1 - 9 = -8 \neq 0$, so discontinuous at $x=1$. (l) $y = \frac{1}{x - 1}$ for $x \in \mathbb{Z}$: Discontinuous at $x=1$. (m) $f(x) = \frac{1}{x^2}$ for $x \in \mathbb{Z}$: Discontinuous at $x=0$ (not in $\mathbb{Z}$ if zero excluded, else discontinuous). (n) $y = |x|$ for $x \in \mathbb{Z}$: Continuous on integers. 5. **Summary:** - Continuous: (a), (b), (c), (d), (e), (f), (g), (k), (l) - Discontinuous: (h), (i), (j) 6. **Desmos Graphs:** - Graphs (a), (b), (c) show discontinuities at open circles as described.