Short on time? This page is the recap. For explanations, the four-panel figure and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → c) f(x) means “the limit as x approaches c of f(x)”; c⁻ and c⁺ mean from the left and from the right.
Recap
- f is continuous at x = c when (1) f(c) exists, (2) lim (x → c) f(x) exists, and (3) lim (x → c) f(x) = f(c).
- Check the conditions in order and stop at the first failure.
- The limit exists only if both one-sided limits exist, are finite and are equal.
- At a point where a piecewise function changes formula, find both one-sided limits, each with its own piece.
- A justification quotes values: f(c) = …, the limit = …, then the comparison.
Key relationships
| Condition | What you show | If it fails, the graph has… |
|---|---|---|
| 1. f(c) exists | A number for f(c), from the right piece or the given value | A hole with no point (removable) or an asymptote |
| 2. lim (x → c) f(x) exists | lim (x → c⁻) f(x) = lim (x → c⁺) f(x), a finite number | A jump, a vertical asymptote or oscillation |
| 3. lim (x → c) f(x) = f(c) | The two numbers from 1 and 2 are equal | A hole with the point placed elsewhere (removable) |
Assumptions behind the method
- Condition 3 compares two numbers, so it is checked only after conditions 1 and 2 hold.
- For f(c), use the piece whose inequality includes c (≤ or ≥), or the separately given value.
- To find the limit you may rewrite the formula (factor, conjugate) as in Topic 1.6, because a limit ignores the value at c.
Mistakes to avoid
- “f(c) exists, so f is continuous.” All three conditions are needed.
- “The limit exists, so f is continuous.” The value at c must match.
- “0/0 in the formula means discontinuous.” If f(c) is given separately and matches the limit, f is continuous.
- Checking one side only at a piecewise boundary.
- Using the wrong piece for f(c). Read ≤ versus <.
- Treating ∞ as a limit that exists. It fails condition 2.
- Justifying with “no break” and no numbers.
Quick self-check
- f(x) = (x² − 25)/(x − 5) for x ≠ 5 and f(5) = 10. Is f continuous at 5? (Yes: f(5) = 10, the limit is 10, and they are equal.)
- f(x) = 2x + 1 for x < 3 and f(x) = x² − 2 for x ≥ 3. Is f continuous at 3? (Yes: f(3) = 7, both one-sided limits are 7.)
- lim (x → c) f(x) = 4 but f(c) is undefined. Which condition fails? (Condition 1.)
- f(c) = 2, lim (x → c⁻) f(x) = 2 and lim (x → c⁺) f(x) = 5. Which condition fails? (Condition 2: the limit does not exist.)
Next: practice questions.