Looking at the provided graph of $y = f(x)$, we need to assess the continuity at $x = 6$.

For a function to be continuous at a point $x = c$, the following three conditions must be met:

1. $f(c)$ is defined.
2. $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$ (the left-hand limit equals the right-hand limit).
3. The limit equals the function value: $\lim_{x \to c} f(x) = f(c)$.

At $x = 6$:

• $f(6)$ appears to be defined as a solid point, located at around $y = 4$.
• The left-hand limit $\lim_{x \to 6^-} f(x)$ approaches $y = 7$ as $x$ approaches 6 from the left.
• The right-hand limit $\lim_{x \to 6^+} f(x)$ also approaches $y = 7$ as $x$ approaches 6 from the right.

Thus, the limit exists and is equal to 7, but $f(6) \neq 7$. This breaks condition 3.

Correct Explanation:

The correct statement is:

$\lim_{x \to 6} f(x) \neq f(6)$

Would you like further clarification on this? Here are 5 related questions to expand your understanding:

1. How can we formally define continuity for any function at a point?
2. What are other examples where a function is discontinuous because $\lim f(x) \neq f(c)$?
3. What happens to a function’s graph when the left-hand and right-hand limits don't match at a point?
4. What types of discontinuities exist, and how do they differ?
5. How does the behavior of limits help in identifying points of discontinuity?

Tip: Always remember, continuity issues at a point can arise due to limits or undefined values—it's essential to check both!