To determine which of the statements are incorrect, we will analyze each statement one by one.
### Statement 1:
**If \( f + g \) is continuous at \( x = a \), then \( f \) and \( g \) are continuous at \( x = a \).**
**Analysis:**
- We can construct functions \( f(x) \) and \( g(x) \) such that \( f + g \) is continuous at a point, but \( f \) and \( g \) themselves are not continuous.
- For example, let:
- \( f(x) = 1 \) for \( x \geq 0 \) and \( f(x) = -1 \) for \( x < 0 \)
- \( g(x) = -1 \) for \( x \geq 0 \) and \( g(x) = 1 \) for \( x < 0 \)
- Then \( f + g = 0 \) for all \( x \), which is continuous everywhere, including at \( x = 0 \).
- However, both \( f \) and \( g \) are discontinuous at \( x = 0 \).
**Conclusion:** This statement is **incorrect**.
### Statement 2:
**If \( \lim_{x \to a} f(x) g(x) \) exists, then both \( \lim_{x \to a} f(x) \) and \( \lim_{x \to a} g(x) \) must exist.**
**Analysis:**
- Consider \( f(x) = x \) and \( g(x) = \frac{1}{x} \).
- As \( x \to 0 \), \( f(x) g(x) = x \cdot \frac{1}{x} = 1 \), which has a limit of 1.
- However, \( \lim_{x \to 0} f(x) \) exists (it approaches 0), but \( \lim_{x \to 0} g(x) \) does not exist (it approaches \( \infty \)).
**Conclusion:** This statement is **incorrect**.
### Statement 3:
**Discontinuity at \( x = a \) implies non-existence of limit at that point.**
**Analysis:**
- This statement is false because there can be removable discontinuities where the limit exists but the function is not continuous.
- For example, if \( f(x) = \frac{x^2 - 1}{x - 1} \) for \( x \neq 1 \) and \( f(1) = 0 \), the limit as \( x \to 1 \) exists and equals 2, but \( f(1) \neq 2 \), indicating a removable discontinuity.
**Conclusion:** This statement is **incorrect**.
### Statement 4:
**All functions defined on a closed interval attain a maximum or minimum value in that interval.**
**Analysis:**
- This statement is true for continuous functions on closed intervals due to the Extreme Value Theorem.
- However, if we consider the function \( f(x) = \tan(x) \) defined on the interval \( (-\frac{\pi}{2}, \frac{\pi}{2}) \), it does not attain a maximum or minimum value as it approaches \( \infty \) and \( -\infty \) at the endpoints.
**Conclusion:** This statement is **incorrect**.
### Summary of Incorrect Statements:
1. Statement 1 is incorrect.
2. Statement 2 is incorrect.
3. Statement 3 is incorrect.
4. Statement 4 is incorrect.
### Final Answer:
All statements (1, 2, 3, and 4) are incorrect.
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