Confirming Continuity Over an Interval

A function continuous over an interval is one you could graph across that whole stretch without lifting your pencil — no jumps, no holes, no asymptotes anywhere inside it. Checking this at infinitely many points one by one is impossible, so instead you lean on a shortcut: polynomials, √x, sin(x), cos(x), and exponentials are continuous everywhere they're defined, and sums, products, quotients, and compositions of continuous functions stay continuous wherever that combination is defined. So confirming continuity over an interval really means finding the natural domain of the function and checking that the interval in question avoids every excluded point.

Take √(x - 3)/(x - 7). The square root needs x - 3 ≥ 0, so x ≥ 3. The denominator can't be zero, so x cannot equal 7. Combining those restrictions, the function is continuous on [3, 7) and on (7, infinity), and the largest interval containing numbers near 3 is [3, 7). Notice x = 3 is included with a one-sided limit — the function only needs to approach its value from the right there, since nothing exists to its left.

This is why checking just the endpoints of an interval is never enough. A function can be perfectly continuous at x = 0 and at x = 5 while hiding a vertical asymptote at x = 2 — and that single interior break destroys continuity on [0, 5] entirely. This idea sets up everything after it: the Intermediate Value Theorem and Extreme Value Theorem both require continuity on a closed interval, and that requirement is exactly this — no breaks anywhere inside, not just good behavior at the ends.