From the image provided, let's analyze the problem step by step:

Problem Overview:

The question asks us to determine the values of $x$ at which the function (represented by the graph) is not continuous. If the function is not continuous, we need to determine why.

Key Definitions:

1. Continuous Function: A function is continuous at a point $x = c$ if:

   • $f(c)$ is defined.
   • The limit as $x \to c$ exists.
   • The limit as $x \to c$ equals $f(c)$.

2. Types of Discontinuities:

   • Point Discontinuity: $f(c)$ is undefined or the limit does not exist.
   • Jump Discontinuity: The left-hand limit ($\lim_{x \to c^-}$) does not equal the right-hand limit ($\lim_{x \to c^+}$).
   • Infinite Discontinuity: The function approaches infinity at $x = c$.

Observing the Graph:

The graph appears to be a continuous curve with no breaks, jumps, or holes. This suggests the function might be continuous for all values of $x$.

Answer Analysis:

• Option A: "The function is not continuous at $x = \ldots$" – This would require identifying specific points of discontinuity, but none are evident from the graph.
• Option B: "The function is continuous for all values of $x$" – This seems correct because there are no apparent discontinuities.
• Option C: "The function is not continuous at $x = \ldots$" (with a large change in $f(x)$) – Again, no such points are visible.

Correct Answer:

Option B: The function is continuous for all values of $x$.

Would you like further clarification or more detailed steps?

Here are 5 related questions to deepen understanding:

1. What are the mathematical conditions for continuity at a point?
2. How can we differentiate between jump and infinite discontinuities in graphs?
3. What is the importance of left-hand and right-hand limits in determining continuity?
4. Could a graph appear smooth but still have discontinuities? Why?
5. How does the Intermediate Value Theorem relate to continuity?

Tip: When analyzing continuity, always verify the three conditions (existence, limit, and equality of the limit and value).