A function is continuous at a if you can draw it through that point without lifting the pencil.
Quick check
f(x) = (x² − 1)/(x − 1) for x ≠ 1, and f(1) = 5. Is f continuous at x = 1?
Continuity at a
This single equation packs in all three conditions: f(a) exists, the limit exists, and they're equal.
Types of discontinuity
- Removable — a hole; the limit exists but doesn't match (or isn't defined) at the point.
- Jump — left and right limits exist but differ.
- Infinite — the function blows up (a vertical asymptote).
Your turn
Where is f(x) = 1/(x − 3) discontinuous, and what kind?
Three conditions
- f(a) is defined.
- lim(x→a) f(x) exists.
- lim(x→a) f(x) = f(a).