Informally, a function is continuous if you can draw it without lifting your pen. We can make that precise at a single point x = c. A function f is continuous at c when all three of these hold:

1. f(c) is defined — the point exists.
2. \displaystyle\lim_{x \to c} f(x) exists — the curve heads somewhere (both sides agree).
3. They match: \displaystyle\lim_{x \to c} f(x) = f(c).

That third condition is the punchline: where the curve is heading is exactly where the point is. This is precisely why direct substitution works — it works exactly when the function is continuous.

The three-part test, live

Flip between four cases below. Only one passes all three checks; watch which condition breaks in each of the others.

Three ways to break

When a function isn't continuous at c, the type of discontinuity tells you which condition failed — built on when a limit doesn't exist.

• Removable (a hole) — the limit exists, but f(c) is missing or sits at the wrong height. You could "remove" it by redefining one point.
• Jump — the two one-sided limits disagree, so the limit DNE. No single value could patch it.
• Infinite — a vertical asymptote; the function blows up and the limit DNE.

A worked check

Is f(x) = x^2 + 1 continuous at x = 2?

• f(2) = 5 — defined. ✓
• \lim_{x \to 2}(x^2 + 1) = 5 — exists. ✓
• They match: 5 = 5. ✓

All three pass, so f is continuous at 2 — pen never lifts. Every polynomial is continuous everywhere, which is why substitution never fails for them.

Watch Sal define continuity