Let's analyze the given graph of the function $f$ and evaluate the truth of the three statements.

1. Statement I:

$\lim_{{x \to 2}} f(x) = f(2)$

• Looking at the graph near $x = 2$, the value of $f(2)$ is indicated by a closed dot, showing $f(2) = 0$.
• To check the limit $\lim_{{x \to 2}} f(x)$, we observe the behavior of the function as $x$ approaches 2 from both sides.
  • As $x \to 2^-$, the graph approaches 0.
  • As $x \to 2^+$, the graph also approaches 0.
  • Therefore, $\lim_{{x \to 2}} f(x) = 0$.

Since both the limit and the function value at $x = 2$ are equal, Statement I is true.

2. Statement II:

$\lim_{{x \to 6^-}} f(x) = \lim_{{x \to 6^+}} f(x)$

• From the graph:
  • As $x \to 6^-$, the function approaches the value $f(x) = 3$.
  • As $x \to 6^+$, the function approaches $f(x) = -3$.

Since the left-hand limit $\lim_{{x \to 6^-}} f(x) = 3$ and the right-hand limit $\lim_{{x \to 6^+}} f(x) = -3$ are not equal, Statement II is false.

3. Statement III:

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

• From the graph:
  • $f(6)$ is given by a closed dot at $(6, 3)$, so $f(6) = 3$.
  • As analyzed above, $\lim_{{x \to 6^-}} f(x) = 3$ and $\lim_{{x \to 6^+}} f(x) = -3$, meaning that $\lim_{{x \to 6}} f(x)$ does not exist.

Since the limit does not exist, Statement III is false.

Conclusion:

• Statement I is true.
• Statements II and III are false.

Would you like more details or explanations?

Here are some related questions you can explore:

1. What is the formal definition of a limit?
2. How do left-hand and right-hand limits affect continuity?
3. What are some key conditions for a function to be continuous at a point?
4. What is the significance of open and closed dots in function graphs?
5. How can you identify jump discontinuities in a graph?

Tip: Always check the behavior of the function from both sides when calculating limits at a given point.