Continuous function

Continuity at a point is a local test: $\lim_{x \to a} f(x) = f(a)$ at that one value of $a$. Calling $f$ a continuous function is a global claim, made once the local test passes at every point of the domain. Exam wording matters here, since a question asking whether $f$ is continuous at $x = 3$ is answered with three checks, while a question asking whether $f$ is continuous on $[1, 4]$ requires the property to hold across the whole interval, with one-sided limits at the two endpoints.

You almost never verify continuity from scratch. Polynomials, $\sin x$, $\cos x$, and $e^x$ are continuous on all real numbers, and sums, differences, products, quotients, and compositions of continuous functions are continuous wherever the result is defined. That closure property is what lets direct substitution evaluate most limits in Unit 1, and it is why continuity is worth checking before quoting the Intermediate Value Theorem, the Extreme Value Theorem, or the Mean Value Theorem.