Confirming Continuity over an Interval AB & BC

Continuity on an interval means continuity at every point of it, so a single point proves nothing. The workable argument names the family the function belongs to, since polynomials are continuous everywhere and rational, radical, exponential, logarithmic and trigonometric functions are continuous on their domains, and then checks the finitely many suspect points: denominator zeros, domain edges, and piecewise joins. At the endpoints of a closed interval, continuity is one-sided, because inputs outside the interval are not part of the claim.

Two errors dominate. The first is generalizing from one point, verifying continuity at a convenient input and asserting it across the whole interval, which misses an asymptote or a failed join sitting somewhere in the middle. The second is applying the three-part test incompletely, and in particular demanding a two-sided limit at a closed endpoint, which would wrongly disqualify a function such as $\sqrt{x}$ on $[0, 4]$.