Let's analyze the three statements in the question:

1. Statement I:
   "If $f$ is defined at $x = a$, then $\lim_{x \to a} f(x) = f(a)$."

   • This statement is not necessarily true. A function being defined at $x = a$ does not imply that the limit as $x \to a$ equals $f(a)$. There could be a discontinuity at $x = a$. So, Statement I is false.

2. Statement II:
   "If $f$ is continuous at $x = a$, then $\lim_{x \to a} f(x) = f(a)$."

   • This is the definition of continuity. If a function is continuous at a point $a$, then the limit as $x \to a$ of $f(x)$ is equal to $f(a)$. So, Statement II is true.

3. Statement III:
   "If $f$ is differentiable at $x = a$, then $\lim_{x \to a} f(x) = f(a)$."

   • Differentiability at a point implies continuity at that point. Since a differentiable function is always continuous, this means the limit as $x \to a$ of $f(x)$ equals $f(a)$. So, Statement III is true.

Conclusion:

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

Thus, the correct answer is C (II and III only).

Would you like any further details or clarifications?

Here are 5 related questions to consider:

1. What is the formal definition of a limit?
2. Can a function be continuous but not differentiable?
3. How does differentiability imply continuity?
4. Are there any cases where a limit exists but the function is not defined at that point?
5. What are examples of functions that are differentiable everywhere?

Tip: A differentiable function is always continuous, but a continuous function is not always differentiable!