Short on time? This page is the recap. For explanations, the diagrams and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → c⁻) and lim (x → c⁺) are the limits from the left and from the right.
Recap
- f is continuous on an interval if it is continuous at every point of the interval.
- Open interval (a, b): the usual two-sided definition at every point inside.
- Closed interval [a, b]: continuous on (a, b), plus lim (x → a⁺) f(x) = f(a) and lim (x → b⁻) f(x) = f(b).
- Polynomial, rational, power, exponential, logarithmic and trig functions are continuous at every point of their domains.
- Sums, differences, products, quotients (denominator ≠ 0) and composites of continuous functions are continuous.
Key relationships
| Function | Continuous on |
|---|---|
| Polynomial | (−∞, ∞) |
| Rational p(x)/q(x) | all x with q(x) ≠ 0 |
| √x (even root) | [0, ∞) |
| ∛x (odd root) | (−∞, ∞) |
| eˣ, bˣ (b > 0) | (−∞, ∞) |
| ln x | (0, ∞) |
| sin x, cos x | (−∞, ∞) |
| tan x, sec x | all x except odd multiples of π/2 |
| cot x, csc x | all x except multiples of π |
Piecewise boundary c: continuous at c only if left limit = right limit = f(c). If only one side matches f(c), c joins the interval on that side.
Assumptions behind the method
- “Continuous on its domain” is not the same as “continuous on every interval”. An interval must contain no excluded value.
- Each piece of a piecewise function is judged only on the x values it actually uses.
- At an endpoint of a closed interval, only the one-sided limit from inside the interval is needed.
Mistakes to avoid
- Calling 1/x “continuous on [−1, 1]”. Its domain has a gap at 0.
- Missing a restriction: check every denominator, even root and logarithm.
- Testing only one side at a piecewise boundary.
- Wrong brackets: square only if the value exists and the one-sided limit matches it.
- Worrying about an asymptote outside a piece’s own interval.
- Forgetting removable discontinuities: a hole still breaks continuity, even though the limit exists.
Quick self-check
- Where is ln(x − 3) continuous? ((3, ∞): a logarithm needs x − 3 > 0)
- Where is (x + 1)/(x² − 9) continuous? ((−∞, −3), (−3, 3) and (3, ∞))
- f(x) = x² for x ≤ 2 and f(x) = 2x for x > 2. Is f continuous on (−∞, ∞)? (Yes. Both pieces are polynomials, and at x = 2 the left limit, right limit and f(2) are all 4.)
- Is tan x continuous on [0, 1]? (Yes. The first positive value it misses is π/2 ≈ 1.57, which is outside [0, 1].)
Next: practice questions.