What is it
A function is continuous at a point if its limit there equals its actual value — informally, if you can draw its graph through that point without lifting your pen — and continuity is a prerequisite for many of calculus's most important theorems.
Why it matters
Continuity is the assumption behind major results like the Intermediate Value Theorem and the Extreme Value Theorem — without it, a function can behave unpredictably, jumping or breaking in ways that make standard calculus techniques invalid.
Exam tip
To prove continuity formally, check all three conditions explicitly: the function is defined at the point, the limit exists at the point, and the limit equals the function's value there — skipping any one of the three is an incomplete proof.
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