Confirming continuity over an interval means analyzing a function to ensure it is continuous at every single point within that interval.
This is not just about checking individual points; it's about understanding the behavior of the function across an entire stretch.
The mechanism you use is the definition of continuity: a function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point.
This needs to hold true for every point in the interval.
The challenge comes when dealing with functions that have complex domains or potential discontinuities, such as rational functions with asymptotes or removable discontinuities.
You might think a function is continuous because it looks smooth on a graph, but without checking the limits and values explicitly, you could miss hidden discontinuities.
Misjudging continuity can lead to incorrect conclusions in later calculus problems, particularly when applying theorems that assume continuity over an interval.