Theorem. If a function Fx is continuous from x=a to x=b inclusive, then among all the values which it takes, if we imagine that x successively takes all the values from a to b inclusive, there is always a greatest in the sense that no other is greater than it, and there is also a smallest in the sense that no other is smaller than it.
Dai manoscritti di Bernard Bolzano (Functionenlehre e Größenlehre 1817)
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