WHY WE DIDNOT STUDY THE CONTINUOS AT X=2 

We didn't study the continuity at \( x = 2 \) because \( x = 2 \) is the right endpoint of the function's domain. The concept of continuity typically concerns the behavior of the function as \( x \) approaches a point from both sides. 

### Key Points:

- **Continuity Definition**: For a function to be continuous at a point \( x = c \), the following conditions must be met:
  1. \( f(c) \) is defined.
  2. The limit \( \lim_{x \to c} f(x) \) exists.
  3. The limit \( \lim_{x \to c} f(x) \) equals \( f(c) \).

- **At \( x = 2 \)**: The function is defined only up to \( x = 2 \). There's no need to check the continuity at \( x = 2 \) from the right because the function doesn't extend beyond \( x = 2 \). We only need to check the left-hand limit:
  - As \( x \) approaches 2 from the left, the function is \( f(x) = mx + b \), so \( \lim_{x \to 2^-} f(x) = m(2) + b \).
  - Since this is the endpoint, the function is continuous at \( x = 2 \) if it's defined at this point, which it is because \( f(2) = 2m + b \) is a valid value.

Thus, there is no requirement to study the continuity at \( x = 2 \) further.

Do you want more details, or do you have any questions?

Here are some related questions:
1. How would the analysis change if the domain extended beyond \( x = 2 \)?
2. What does it mean for a function to be continuous at a boundary point?
3. What if the function were undefined at \( x = 2 \)? 
4. How does the continuity at \( x = 2 \) affect the overall behavior of the function?
5. Can a function be continuous at only one side of a boundary point?
6. How would we test for continuity if the function included \( x > 2 \)?
7. What are the implications of piecewise-defined functions at boundary points?
8. How does the concept of left-hand and right-hand limits help in analyzing functions?

**Tip:** When dealing with endpoints of a domain, consider only the one-sided limit that applies since the function isn't defined beyond the endpoint.

Explore why continuity is not studied at x = 2 in function domains. Learn about the definition of continuity, limits, and how they apply to functions at domain endpoints. Suitable for undergraduate students and anyone interested in mathematical analysis.

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