1.11 Defining Continuity at a Point

Three conditions

f is continuous at c when all three hold: (1) f(c) is defined, (2) lim(x→c) f(x) exists, and (3) lim(x→c) f(x) = f(c).

If any fail

If even one condition fails, f has a discontinuity at c. Note continuity does NOT require the derivative to exist — that's a stronger property.

Key terms

Continuity at a point

f is continuous at c if three conditions all hold.

Condition 1

f(c) is defined.

Condition 2

lim(x→c) f(x) exists.

Condition 3

lim(x→c) f(x) = f(c).

Not required

The derivative need not exist for continuity.

Try an AP-style question

Which is NOT one of the three conditions for f to be continuous at x = a?

1. lim(x→a) f(x) = f(a)
2. lim(x→a) f(x) exists
3. f'(a) exists
4. f(a) is defined