Continuity & the Intermediate Value Theorem
A function is continuous at x = c when the limit at c exists and equals f(c). The Intermediate Value Theorem uses continuity to guarantee a function hits every value between two outputs.
The three-part definition of continuity
f is continuous at x = c exactly 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 one fails, f is discontinuous at c.
Discontinuities come in types: removable (a hole — the limit exists but doesn't match f(c), or f(c) is undefined), jump (the one-sided limits exist but disagree), and infinite/essential (the function grows unboundedly near c).
Intermediate Value Theorem (IVT)
If f is continuous on the closed interval [a, b], then for any value M between f(a) and f(b), there is at least one c in [a, b] with f(c) = M. Intuitively: a continuous curve can't skip over a height without passing through it.
IVT is an existence statement — it guarantees a c exists, but doesn't tell you its exact value or how many such c's there are.
Worked Example
Show f(x) = x³ - x - 2 has a root between x = 1 and x = 2.
- f is a polynomial, so it's continuous everywhere, including on [1,2].
- f(1) = 1 - 1 - 2 = -2. f(2) = 8 - 2 - 2 = 4.
- Since f(1) = -2 < 0 < 4 = f(2), and 0 is between f(1) and f(2), IVT guarantees some c in (1,2) with f(c) = 0.
Answer: By IVT, a root exists in (1, 2).
Key terms (2)
- Removable discontinuity
- A point where the limit exists but doesn't equal f(c) (or f(c) is undefined) — visually, a hole.
- Intermediate Value Theorem
- A continuous function on [a,b] takes on every value between f(a) and f(b) at least once.
Practice Quiz
f(x) = (x²-1)/(x-1) for x ≠ 1, and f(1) = 2. Is f continuous at x = 1?