Il teorema degli zeri - Bernard Bolzano (1817)

Purely Analytic

Proof of the Theorem

that

between any two Values, which give Results of Opposite Sign,

there lies at least one real Root of the Equation

by

Bernard Bolzano

Priest, Doctor of Philosophy, Professor of Theology and Ordinary Member of the Royal Society of Sciences at Prague

For the Proceedings of the Royal Society of Sciences

Prague, 1817 Printed by Gottlieb Haase

Theorem. If two functions of x, fx and φx, vary according to the law of continuity either for all values of x or for all those lying between α and β, and furthermore if fα < φα and fβ > φβ, then there is always a certain value of x between α and β for which fx = φx.