To determine whether the function $f$ is continuous on its domain, we need to check for any breaks, jumps, or undefined points within the domain of the function. A function is continuous on its domain if, for every point $c$ in the domain:

1. The function $f(c)$ is defined.
2. The limit $\lim_{x \to c} f(x)$ exists.
3. The limit equals the function value, i.e., $\lim_{x \to c} f(x) = f(c)$.

Analysis of the Graph:

• The graph appears smooth except at the $y$-intercept.
• At the $y$-axis ($x = 0$), there is an open circle at $y = 1$, indicating that the function is not defined at $x = 0$.
• Since there is no filled dot at $x = 0$, $f(0)$ is undefined.

Continuity Conclusion:

The function is not continuous on its domain because it is not defined at $x = 0$.

Correct Answer:

• The function is not continuous on its domain.
• The correct explanation is: "The function is not defined at the end points."

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Do you want more details or have any questions?

5 Related Questions:

1. What does it mean for a function to be continuous at a specific point?
2. How do open and closed circles on a graph affect continuity?
3. What are the types of discontinuities (e.g., jump, infinite, removable)?
4. How do you determine the domain of a function from its graph?
5. What is the relationship between limits and continuity?

Tip:

Always check for open and closed points on the graph, as they indicate whether a function is defined at that specific $x$-value.