A continuous function is a function where small changes in the input lead to small changes in the output. In other words, there are no sudden jumps, breaks, or gaps in the graph of the function. Formally, a function $f(x)$ is continuous at a point $x = a$ if the following three conditions are satisfied:

1. The function is defined at $x = a$: $f(a)$ exists.
2. The limit of the function as $x$ approaches $a$ exists: $\lim_{x \to a} f(x)$ exists.
3. The value of the function at $a$ equals the limit of the function as $x$ approaches $a$: $\lim_{x \to a} f(x) = f(a)$.

Formal Definition of Continuity:

A function $f(x)$ is continuous on an interval if it is continuous at every point in that interval. This means:

$\lim_{x \to c} f(x) = f(c)$

for all $c$ in the interval.

Key Examples:

1. Polynomial functions (e.g., $f(x) = x^2 + 3x + 2$) are continuous everywhere.
2. Rational functions (e.g., $f(x) = \frac{x}{x-1}$) are continuous everywhere except where the denominator is zero (at $x = 1$, in this case).
3. Trigonometric functions (e.g., $\sin(x), \cos(x)$) are continuous on their entire domains.

Continuity on Intervals:

• Continuous on an open interval: A function is continuous on an open interval $(a, b)$ if it is continuous at every point inside the interval.
• Continuous on a closed interval: A function is continuous on a closed interval $[a, b]$ if it is continuous on $(a, b)$, and the limits at the endpoints $a$ and $b$ equal the function values at those points.

Intuitive Idea:

• You can draw the graph of a continuous function without lifting your pen.

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Would you like more detailed examples or a specific explanation of certain types of functions?

Here are 5 questions to further explore continuous functions:

1. What is the difference between continuous and discontinuous functions?
2. How do you determine the points of discontinuity in a function?
3. Can a piecewise function be continuous? If so, how?
4. What is the significance of continuity in calculus, especially with respect to integration and differentiation?
5. How do continuous functions behave on closed intervals in the context of the Intermediate Value Theorem?

Tip: A continuous function on a closed interval always attains both a maximum and a minimum value (Extreme Value Theorem).