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

Verified step by step guidance

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.

Key Concepts

Continuity of Functions

A function is said to be continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph of the function at that point. For a function to be continuous on an interval, it must be continuous at every point within that interval.

Types of Intervals

An interval is a range of values on the real number line, which can be open, closed, or half-open. A closed interval includes its endpoints, while an open interval does not. Understanding the type of interval is crucial because it affects whether the endpoints are included in the continuity assessment of the function.

Limit of a Function

The limit of a function describes the behavior of the function as it approaches a particular point from either side. For continuity, the left-hand limit and right-hand limit at a point must exist and be equal to the function's value at that point. This concept is foundational in calculus, as it helps determine the smoothness and predictability of functions.