Interval Notation and Continuity of Functions

Interval Notation and the Nature of Points Within

Interval notation is a mathematical shorthand used to describe the set of numbers lying between two endpoints, which may or may not be included in the set. An interval can be open, closed, or half-open (also known as half-closed). In interval notation, square brackets [a, b] denote a closed interval, including the endpoints a and b, while parentheses (a, b) indicate an open interval, excluding the endpoints. The notation [a, b) or (a, b] represents half-open intervals, including only one of the endpoints. Inequality notation can also express these intervals, for example, \( a \le x \le b \) for a closed interval and \( a < x < b \) for an open interval. The set of all real numbers is denoted by \( (-\infty, \infty) \) or simply \( \mathbb{R} \). Within an interval, points that are not endpoints are referred to as interior points.

Defining Continuity Over an Interval

A function is said to be continuous over an interval if it does not have any interruptions or jumps in its graph within that interval. For a function to be continuous on an interval, it must be continuous at every interior point and, if the interval includes its endpoints, the function must also be continuous from the right at the left endpoint and from the left at the right endpoint. This means that the limit of the function as it approaches any point within the interval from either direction must equal the function's value at that point. For closed intervals, this includes the limits as the variable approaches the endpoints.

Steps to Determine Continuity on an Interval

To determine if a function is continuous on a given interval, one must first ensure that the interval lies within the domain of the function. Then, one must check that the function is continuous at every interior point of the interval by verifying that the limit exists and equals the function's value at those points. If the interval includes its endpoints, the continuity of the function at these points must be assessed from the appropriate direction. This systematic approach is essential for establishing the continuity of a function over any interval.

Examples Illustrating Continuity Over Intervals

Consider the function \( f(x) = \sqrt{x - 4} \), which is defined for \( x \ge 4 \). When examining the interval [0, 7), the function is not continuous since the interval includes values for which the function is not defined. However, on the interval [4, 7), the function is continuous because it satisfies the criteria for continuity: it is defined and continuous at all interior points, and it is continuous from the right at the left endpoint, x = 4. Graphical analysis can also be a helpful tool in determining continuity, as a continuous function will have an unbroken graph over the interval.

Theorems Related to Continuity Over Intervals

There are several theorems that provide insights into the continuity of functions over intervals. For example, the theorem that all polynomial functions are continuous over the entire set of real numbers (\( \mathbb{R} \)) is particularly useful. This is because polynomials do not have breaks, jumps, or points of discontinuity. Similarly, rational functions, which are ratios of polynomials, are continuous wherever they are defined; their discontinuities only occur at points where the denominator is zero. These theorems are valuable as they allow for a general understanding of the continuity of these classes of functions without the need for point-by-point analysis.

Key Takeaways on Continuity Over an Interval

In conclusion, a function is continuous on an interval if it is continuous at all points within that interval, including being continuous from the correct direction at any included endpoints. A function that is continuous at every point in its domain is described as being continuous on its domain. The specific intervals where a function maintains continuity are referred to as intervals of continuity. Furthermore, a function that is continuous at every point on the real number line is said to be continuous everywhere. Mastery of these concepts is crucial for the study of calculus and mathematical analysis, as they are foundational to understanding the behavior of functions.