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.