What does it mean for a function to be continuous on an interval?

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1. Step 1: Understand the definition of continuity at a point. A function f(x) is continuous at a point x = c if the following three conditions are met: 1) f(c) is defined, 2) the limit of f(x) as x approaches c exists, and 3) the limit of f(x) as x approaches c is equal to f(c).
2. Step 2: Extend the concept of continuity to an interval. A function is continuous on an interval if it is continuous at every point within that interval.
3. Step 3: Consider different types of intervals. The interval can be open (a, b), closed [a, b], or half-open (a, b] or [a, b). The function must be continuous at every point in the interval, including the endpoints if they are part of the interval.
4. Step 4: Visualize continuity. Graphically, a function is continuous on an interval if you can draw the function on that interval without lifting your pencil from the paper.
5. Step 5: Recognize the importance of continuity. Continuity is a fundamental concept in calculus because it ensures the function behaves predictably, allowing for the application of various theorems and techniques, such as the Intermediate Value Theorem and the Fundamental Theorem of Calculus.