Real Analysis 32 | Intermediate Value Theorem [dark version]

The Intermediate Value Theorem (IVT) guarantees that a continuous function cannot “skip” values between its endpoint outputs on an interval. If a function f is continuous on [a, b] and takes values f(a) and f(b), then every number Y between f(a) and f(b) occurs as f(x) for some x in [a, b]. This matters because it turns a geometric intuition—“a continuous graph can be drawn in one stroke”—into a precise mathematical guarantee about what values are achievable.

The proof starts by reducing the problem to a simpler target: finding an x where the function equals a chosen Y. Given any Y between f(a) and f(b), the method defines a new function G(x) = f(x) − Y. Continuity carries over, and the goal becomes finding a point where G(x) = 0. Next comes a second normalization: the proof arranges a sign change across the interval. If G(a) is already positive, the left endpoint can serve as the “positive side” and the right endpoint as the “negative side” (or vice versa). If not, the function is mirrored (implemented by defining a modified function F̃ that flips the sign when needed) so that the setup always becomes: F̃(a) ≤ 0 and F̃(b) ≥ 0, with a continuous function in between.

With the sign change in place, the interval [a, b] is repeatedly bisected. At each step, the midpoint c is chosen, and the sign of F̃(c) determines which half still contains a zero. If F̃(c) > 0, the proof keeps the left subinterval; otherwise it keeps the right subinterval. This produces nested intervals whose endpoints form two sequences (A_n and B_n). By construction, the intervals shrink in length and the left/right sign conditions persist: the function remains ≤ 0 on one side and ≥ 0 on the other.

Completeness of the real numbers then supplies a limit point x̃ in the intersection of these shrinking intervals. The sign information survives the limit because F̃ is continuous: taking the limit of F̃(A_n) and F̃(B_n) yields that F̃(x̃) must be both ≤ 0 and ≥ 0. The only number satisfying both inequalities is 0, so F̃(x̃) = 0. Finally, the proof translates back to the original function. Since F̃ was defined from G by possibly flipping signs, F̃(x̃) = 0 implies G(x̃) = 0, meaning f(x̃) − Y = 0 and therefore f(x̃) = Y.

A corollary follows immediately: the image of a continuous function on [a, b] is itself an interval, running from the minimum value of f to its maximum value. In short, continuity forces the function to hit everything in between—no gaps, no jumps, and no escape from the range between endpoints.