Confirming Continuity over an Interval

Inside an interval, continuity at a point c means the limit as x approaches c from both sides equals f(c). But at an endpoint, there's only one side to approach from — you can't come from the left of the interval's left endpoint, because that's outside the interval entirely. So at a closed left endpoint a, you only need the right-hand limit to match f(a). At a closed right endpoint b, you only need the left-hand limit to match f(b). This is why endpoints get a different, one-sided check: the function doesn't need to behave on the side that isn't part of the interval.

To confirm continuity over an entire interval, you're really doing two jobs. First, rule out any x-values inside the interval where the function breaks — usually where a denominator hits zero, a piecewise definition jumps, or a root goes negative. Second, check that at each included endpoint, the one-sided limit from inside the interval lands exactly on the function's value there. If an endpoint is open, you skip that check altogether since the function doesn't need to be defined there at all.

This distinction matters because 'continuous on its domain' and 'continuous everywhere' are not the same claim. A function can be perfectly continuous at every point it's defined and still have gaps in its domain where it simply doesn't exist — those gaps are not violations of continuity, they're just excluded territory. This idea sets up the Extreme Value Theorem: that theorem guarantees a maximum and minimum value only when a function is continuous on a closed interval, endpoints included, which is exactly the check you just learned to do.