StellarA function is continuous over an interval if it is continuous at every point in that interval. For an open interval, this means the function must satisfy the continuity conditions everywhere inside the interval. For a closed interval [a,b], the function must also be right-continuous at a and left-continuous at b. In other words, endpoints are checked with one-sided limits because there is no interval beyond them inside the domain.
This lesson extends the idea of continuity from a single point to an entire stretch of the graph. Many common functions are continuous on their domains, including polynomials everywhere and rational functions wherever their denominators are not zero. When students identify continuity over intervals, they are preparing for major theorems later in calculus, including the Intermediate Value Theorem and results about extrema, all of which require continuity assumptions on intervals.