intermediate value theorem

If you climb a hill continuously from a low point to a high point, you must pass through every altitude in between — you cannot teleport over an elevation. The intermediate value theorem makes this exact: a continuous function on an interval skips no value between any two it attains.

Precisely: if f is continuous on a closed interval [a, b] and y is any number strictly between f(a) and f(b), then there exists c in (a, b) with f(c) = y. The conclusion rests squarely on continuity together with the completeness of the real numbers — over the rationals it fails, because the line has gaps to slip through.

Two warnings. The theorem guarantees existence of c, not uniqueness — there may be many. And it is a one-way street: a function can attain all intermediate values yet be discontinuous (this ‘Darboux property’ is genuinely weaker than continuity). A famous application: any odd-degree real polynomial has a real root, since it runs from negative to positive infinity and must cross zero.