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The problem presented is as follows:

Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a continuous function. Show that the set \( A = \{x \in \mathbb{R} : f(x) = 2\} \) is a closed set.

### Proof:

To prove that the set \( A = \{x \in \mathbb{R} : f(x) = 2\} \) is closed, we need to show that the complement of \( A \) in \( \mathbb{R} \) is open. 

1. **Definition of a Closed Set:**
   A set \( A \subset \mathbb{R} \) is closed if its complement, \( A^c = \mathbb{R} \setminus A \), is open.

2. **Complement of \( A \):**
   The complement of \( A \) is 
   \[
   A^c = \{x \in \mathbb{R} : f(x) \neq 2\}.
   \]

3. **Openness of the Complement \( A^c \):**
   Since \( f \) is continuous, for every point \( x_0 \in A^c \), where \( f(x_0) \neq 2 \), there exists an open interval \( (x_0 - \delta, x_0 + \delta) \) around \( x_0 \) such that for all \( x \) in this interval, \( f(x) \neq 2 \). This is because the preimage of an open set under a continuous function is open, and the set \( \{y \in \mathbb{R} : y \neq 2\} \) is open in \( \mathbb{R} \).

   Therefore, \( A^c \) is open in \( \mathbb{R} \).

4. **Conclusion:**
   Since \( A^c \) is open, \( A \) must be closed.

Thus, the set \( A = \{x \in \mathbb{R} : f(x) = 2\} \) is closed.

Would you like more details or have any further questions? 

### Further Questions:
1. What is an example of a continuous function where the set \( A \) is non-empty and closed?
2. How would the proof change if \( f \) were not continuous?
3. Can you explain the concept of open and closed sets in metric spaces?
4. How would you prove a similar result for a set defined by \( f(x) = c \) for some constant \( c \)?
5. What happens if \( f(x) \) is a discontinuous function? Is \( A \) still closed?
6. How do different topologies on \( \mathbb{R} \) affect the definition of closed sets?
7. Can you describe the relationship between continuous functions and closed sets?
8. What role does the intermediate value theorem play in the context of continuous functions and closed sets?

### Tip:
Remember, in topology, the concept of a closed set is closely related to limits. A set is closed if it contains all its limit points, which is essential in understanding the behavior of functions in analysis.

Explore a detailed proof showing that the set A = {x in R : f(x) = 2} is closed for a continuous function f(x). This problem involves understanding the properties of closed sets and the continuity of functions in topology.

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