A corollary of the bijection and practical applications

The bijection theorem (TVI + strict monotonicity)

If f is continuous and strictly monotonic on [a, b], then for every k strictly between f(a) and f(b), there exists a UNIQUE c in ]a, b[ such that f(c) = k. Continuity ensures existence (TVI), whilst strict monotonicity ensures uniqueness.

Application: existence and uniqueness of a root

Let us consider f(x) = x³ - x - 1 on [1, 2]. We calculate f'(x) = 3x^2 - 1, which is strictly positive on [1, 2] (since x ≥ 1 implies 3x^2 - 1 ≥ 2 > 0), so f is strictly increasing. Given that f(1) = -1 < 0 < 5 = f(2), the bijection theorem guarantees the existence and uniqueness of a c in ]1, 2[ such that f(c) = 0.

The bisection method for finding the root

Step	Interval	Sign of f at the midpoint
0	[1, 2]	f(1.5) = 0.875 > 0
1	[1, 1.5]	f(1.25) = -0.297 < 0
2	[1.25, 1.5]	f(1.375) ≈ 0.224 > 0

We thus narrow the range of c more and more precisely (c ≈ 1.3247, the plastic number).

Classic pitfall

Do not confuse the TVI (which only establishes the EXISTENCE of at least one solution) with the bijection theorem (which establishes both existence AND uniqueness thanks to strict monotonicity). Without the monotonicity assumption, there may be several solutions c in [a, b] such that f(c) = k, and asserting uniqueness without justifying strict monotonicity is a common error in exam papers.