How do you know a function is continuous 

A function is considered **continuous** if it meets the following three conditions at every point in its domain:

1. **The function is defined at the point**: For a function \( f(x) \) to be continuous at \( x = a \), the value \( f(a) \) must exist. In other words, \( a \) must be within the domain of the function.
   
2. **The limit of the function exists at the point**: The limit of the function as \( x \) approaches \( a \), denoted as \( \lim_{{x \to a}} f(x) \), must exist. This means that as \( x \) gets arbitrarily close to \( a \), the values of \( f(x) \) approach some finite number.

3. **The limit equals the function's value at the point**: Finally, the limit of the function as \( x \) approaches \( a \) must be equal to the value of the function at \( a \). In other words:
   \[
   \lim_{{x \to a}} f(x) = f(a)
   \]

If these three conditions hold true for every point in the domain of the function, the function is continuous everywhere in its domain. 

### In summary:
- **Defined** at \( a \)
- **Limit exists** as \( x \to a \)
- **Limit equals the value** at \( a \)

If any of these conditions fail, the function is said to be **discontinuous** at that point.

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### Example:
Consider the function \( f(x) = x^2 \). It's continuous everywhere because:
- \( f(x) \) is defined for all \( x \) (so \( f(a) \) exists for any \( a \)),
- The limit \( \lim_{{x \to a}} x^2 = a^2 \) exists,
- \( \lim_{{x \to a}} f(x) = f(a) = a^2 \).

Do you have any specific questions about this process?

### Related questions:
1. What is an example of a function that is discontinuous at a point?
2. How can you visually detect if a function is continuous from its graph?
3. What is the difference between a removable and a non-removable discontinuity?
4. How do you define continuity for functions of multiple variables?
5. Can a piecewise function be continuous? If yes, how?

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### Tip:
A continuous function has no "jumps," "gaps," or "holes" in its graph. When graphing a continuous function, you should be able to draw it without lifting your pen.

Learn how to determine if a function is continuous by understanding the three key conditions: the function must be defined, the limit must exist, and the limit must equal the function's value. This article provides examples and a clear explanation for students in Grades 11-12 and early college.

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