An existence theorem is a theorem that proves the existence of an entity or entities without telling how many entities there are or how to find them. One example of an existence theorem is that for all continuous polynomials, if a value of the polynomial is positive for one value of x, and negative for another value of x, then the value of the polynomial must be zero somewhere in between the two values of x.
In figure 1, the points (-2.5,0.875) and (-1,-4) are plotted. Since f(-2.5) is positive, and f(-1) is negative, then for some value of x, -2.5 < x < -1, f(x)=0. Notice that this theorem does not tell is for how many values of x that f(x) is zero, nor how to find the value of f(x).
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