Bolzano's Theorem of Existence of Zeroes

For a function continuous over an interval, like we proved that having the same value means it must go up eventually if it's going down or if it's going up it must come up eventually, we can also prove something similar.

This is Bolzano's Theorem of Existence of Zeroes:

If a function continuous over has a different sign in two points of this interval, due to continuity, and the fact that it must take on any value between them, it makes sense that to go from negative to positive or from positive to negative it must cross zero.

A function continuous over an interval, that switches sign across this interval, must cross zero. So, there exists at least one point where this function is zero.

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