Let us analyze the problem and justify the selected options based on the given graph:

1. Limits at $t \to 2^-$ and $t \to 2^+$:

   • As $t$ approaches 2 from the left ($t \to 2^-$), the function $f(t)$ approaches the value 10. This matches the graph where the left-hand limit is 10 at $t = 2$.
   • As $t$ approaches 2 from the right ($t \to 2^+$), the function $f(t)$ approaches 2. This also aligns with the graph, showing the right-hand limit is 2 at $t = 2$.

   Hence, $\lim_{t \to 2^-} f(t) = 10$ and $\lim_{t \to 2^+} f(t) = 2$ are correct.

2. Continuity at $t = 2$:

   • A function is continuous at $t = 2$ if $\lim_{t \to 2} f(t) = f(2)$. Here, the left-hand limit and right-hand limit are not equal ($10 \neq 2$), so the two-sided limit $\lim_{t \to 2} f(t)$ does not exist. Therefore, the function is not continuous at $t = 2$.

   Thus, $f$ is not continuous at $t = 2$.

3. Limits at $t \to 4^-$ and $t \to 4^+$:

   • As $t$ approaches 4 from the left ($t \to 4^-$), the function $f(t)$ approaches a value of 2. However, the limit at $t \to 4^-$ is marked as false because it does not match the expected value of 4.
   • As $t$ approaches 4 from the right ($t \to 4^+$), the function $f(t)$ approaches 4. This is correct as per the graph.

   Therefore, $\lim_{t \to 4^+} f(t) = 4$ is correct.

4. Continuity at $t = 6$:

   • At $t = 6$, the graph shows a straight line segment with no breaks or jumps, indicating that the function is continuous at $t = 6$.

   Thus, $f$ is continuous at $t = 6$.

Summary of Correct Options:

• $\lim_{t \to 2^-} f(t) = 10$ (Correct)
• $\lim_{t \to 2^+} f(t) = 2$ (Correct)
• $\lim_{t \to 4^-} f(t) = 4$ (Incorrect)
• $\lim_{t \to 4^+} f(t) = 4$ (Correct)
• $f$ is continuous at $t = 2$ (Incorrect)
• $f$ is continuous at $t = 6$ (Correct)

Let me know if you'd like further clarification or additional examples!

Here are 5 related questions to explore:

1. How do we calculate one-sided and two-sided limits algebraically?
2. Can a function have a left-hand limit and right-hand limit but still be discontinuous?
3. What types of discontinuities exist in functions (e.g., jump, removable)?
4. How can a piecewise function be defined to ensure continuity at all points?
5. Why is continuity important in calculus and real-world applications?

Tip: Always check both the left-hand and right-hand limits at a point to determine continuity!