1. **Problem:** Determine if the function $H(x)$ is continuous at $x=2$ given the graph. 2. **Recall the definition of continuity at a point $x=a$: ** A function $f$ is continuous at $x=a$ if: $$\lim_{x \to a} f(x) = f(a)$$ This means the left-hand limit, right-hand limit, and the function value at $a$ must all be equal. 3. **Analyze continuity at $x=2$: ** - From the graph, the function value at $x=2$ is $H(2) = 1$ (solid dot). - The limit from the left as $x \to 2^-$ is $1$ (line approaches $y=1$). - The limit from the right as $x \to 2^+$ is also $1$ (line extends at $y=1$). 4. Since: $$\lim_{x \to 2^-} H(x) = \lim_{x \to 2^+} H(x) = H(2) = 1,$$ $H(x)$ is continuous at $x=2$. **\boxed{\text{H(x) is continuous at } x=2}$} --- **Note:** The user asked multiple questions but per instructions, only the first problem is solved here.