CAL003 · Calculus · Classes 11–12
Continuity
A function is continuous at a point if you can draw through it without lifting the pen: the limit exists and equals the function's value.
no gapContinuity
📐 Formula
f continuous at a ⇔ lim(x→a) f(x) = f(a)
✏️ Worked example
f(x) = x² (x < 2) and kx (x ≥ 2): left limit 4, right value 2k, so f is continuous exactly when k = 2.
🌍 Where it is used
Smooth motion; the intermediate value theorem.
❓ Quick check
Which is NOT continuous at 0?
🏋️ Practice
EasyIs f(x) = 2x + 1 continuous everywhere?
Answer: Yes
MediumChoose k so kx (x ≥ 1) joins x + 1 (x < 1).
Answer: k = 2
ChallengeIs the floor function continuous at 1?
Answer: No