Let's analyze each function's continuity on the interval $[-1, 3]$ based on the given graphs.

1. $y = f(x)$:

• Observation: There is a hole at $x = 2$ and a solid dot at $x = 2$ in a different location, meaning a jump discontinuity at $x = 2$.
• Conclusion: $f(x)$ is not continuous on $[-1, 3]$ because it fails at $x = 2$.

2. $y = g(x)$:

• Observation: There is an open circle (hole) at $x = 2$, and no value of $g(x)$ is defined at $x = 2$. This represents a removable discontinuity.
• Conclusion: $g(x)$ is not continuous on $[-1, 3]$ because it fails at $x = 2$.

3. $y = h(x)$:

• Observation: There is a jump discontinuity at $x = 0$ and $x = 1$, where the graph abruptly changes its value.
• Conclusion: $h(x)$ is not continuous on $[-1, 3]$ because it fails at $x = 0$ and $x = 1$.

4. $y = k(x)$:

• Observation: There is an open circle (hole) at $x = 0$ and another at $x = 1$. Since no values are defined for $k(x)$ at these points, they represent removable discontinuities.
• Conclusion: $k(x)$ is not continuous on $[-1, 3]$ because it fails at $x = 0$ and $x = 1$.

Final Results:

Would you like further details or an explanation about continuity types?

Related Questions:

1. How do we define continuity mathematically in terms of limits?
2. What is the difference between a removable and a jump discontinuity?
3. How can a graph be modified to make a discontinuous function continuous?
4. Why is the graph of $k(x)$ considered discontinuous at $x = 0$ and $x = 1$?
5. How would the domain affect the continuity of a function on a closed interval?

Tip:

Always check for open circles, jumps, or missing values in the graph to identify discontinuities visually.