https://file.aiquickdraw.com/mathbot/file/1726210957006y6op70dc.jpg

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!

For any function f, which of the following statements must be true? I. If f is defined at x = a, then lim x→a f(x) = f(a). II. If f is continuous at x = a, then lim x→a f(x) = f(a). III. If f is differentiable at x = a, then lim x→a f(x) = f(a).

Explore which conditions must hold true for a function regarding limits, continuity, and differentiability. Learn the core concepts of calculus to understand how functions behave at specific points, with detailed analysis of key theorems and examples.

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