Understanding interval notation and continuity is crucial in mathematics. Interval notation expresses the range of numbers between two endpoints, which can be open, closed, or half-open. Continuity over an interval means a function has no breaks or jumps within that range. The text discusses how to determine if a function is continuous on an interval and highlights theorems related to continuity, such as the behavior of polynomial and rational functions.
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Interval notation is a shorthand used to describe the set of numbers between two endpoints, which can be open, closed, or half-open
Inequality notation can also express intervals, such as \( a \le x \le b \) for a closed interval and \( a < x < b \) for an open interval
To determine if a function is continuous on an interval, one must first ensure that the interval lies within the domain of the function and then check for continuity at every interior point and, if applicable, at the endpoints
A function is said to be continuous over an interval if it has no 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, it must also be continuous from the correct direction at those points
Theorems, such as the continuity of polynomials and rational functions, provide insights into the continuity of functions over intervals without the need for point-by-point analysis
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Definition of continuous function at a point
A function is continuous at a point if the limit as it approaches the point equals the function's value at that point.
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Continuity on closed intervals
For a function to be continuous on a closed interval, it must be continuous at all interior points and have one-sided continuity at the endpoints.
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One-sided continuity at endpoints
A function has right-continuity at the left endpoint and left-continuity at the right endpoint if the one-sided limits equal the function's values at those points.
