To address whether a function \( F \) is continuous for all \( x \), we need to consider the definition of continuity and then apply it to the function \( F \). A function \( F(x) \) is said to be continuous at a point \( x = a \) if the following three conditions are met:

1. **The function is defined at \( a \)**: \( F(a) \) must exist.
2. **The limit exists at \( a \)**: \( \lim_{x \to a} F(x) \) must exist.
3. **The limit equals the function value**: \( \lim_{x \to a} F(x) = F(a) \).

If these conditions are satisfied for every point in the domain of \( F \), then \( F \) is continuous for all \( x \).

### Steps to Determine if \( F \) is Continuous:

1. **Identify the Domain of \( F \)**: Ensure that \( F(x) \) is defined for all \( x \) you are considering.
2. **Check the Limit**: For each point \( a \) in the domain, verify that \( \lim_{x \to a} F(x) \) exists.
3. **Compare the Limit to the Function Value**: Confirm that \( \lim_{x \to a} F(x) = F(a) \).

### Examples of Common Continuous Functions:

- **Polynomials**: Functions like \( F(x) = x^2 + 3x + 2 \) are continuous for all \( x \).
- **Trigonometric Functions**: \( \sin(x) \) and \( \cos(x) \) are continuous for all \( x \).
- **Exponential and Logarithmic Functions**: \( e^x \) is continuous for all \( x \), while \( \ln(x) \) is continuous only for \( x > 0 \).

### Conclusion:

Without a specific function \( F(x) \) provided, it's challenging to definitively state whether \( F \) is continuous for all \( x \). However, the above criteria can be used to analyze any given function. If \( F(x) \) meets these criteria at every point in its domain, then \( F \) is continuous for all \( x \).

Would you like to provide a specific function \( F(x) \) to check its continuity, or do you have any other questions? 

### Related Questions:

1. What are examples of functions that are not continuous at certain points?
2. How do you check the continuity of a piecewise function?
3. Can a function be continuous everywhere but not differentiable?
4. What is the difference between continuity and differentiability?
5. How does the Intermediate Value Theorem relate to continuous functions?

### Tip:

When analyzing the continuity of a function, start by looking for points where the function might be undefined or where the limit does not match the function value—these are common places where discontinuities occur.

Explore how to determine if a function F(x) is continuous for all x. This guide explains the conditions of continuity, including the definition of continuity, limit existence, and function domain. Suitable for high school students in Grades 10-12.

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