Confirming continuity over an interval involves checking that a function is continuous at every point in that interval.
For a function to be continuous at a point, the limit as you approach that point from either side must equal the function's value at that point.
This process requires you to evaluate limits precisely.
The mechanism here is straightforward: you verify continuity by ensuring that the limit from the left equals the limit from the right and both equal the function's value.
Polynomial functions, for instance, are continuous everywhere in their domain because they have no breaks, jumps, or holes.
But rational functions can have discontinuities where the denominator is zero.
The trap is assuming a function is continuous just because it looks smooth on a graph.
You might miss points where the function is undefined or where the limits don't match the function's value.
Misjudging these points leads to incorrect conclusions about the function's behavior over an interval.