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If a temperature sensor reads −5∘C-5^\circ\text{C} at midnight and +5∘C+5^\circ\text{C} at noon, you know the temperature crossed 0∘C0^\circ\text{C} at some point. But in numerical computing, blind sign changes can fool your algorithms into finding 'roots' that do not exist or missing real roots entirely.
The Intermediate Value Theorem (IVT) states that if a function f(x)f(x) is continuous on [a,b][a, b] and f(a)⋅f(b)<0f(a) \cdot f(b) < 0, then there must be at least one value c∈(a,b)c \in (a, b) where f(c)=0f(c) = 0.
In 1817, Czech mathematician Bernard Bolzano proved this property rigorously to eliminate intuition and prove exactly when functions are guaranteed to cross zero without jumping.
What happens if a computer tests two points where signs differ, but the graph jumps across the axis?

False Positives: Vertical Asymptotes

A vertical asymptote is a vertical line x=cx = c where a function grows toward positive or negative infinity as xx approaches cc.
Consider f(x)=1x−2f(x) = \frac{1}{x-2} on the interval [1,3][1, 3]. Evaluating the endpoints gives f(1)=−1f(1) = -1 and f(3)=1f(3) = 1, so f(1)⋅f(3)=−1<0f(1) \cdot f(3) = -1 < 0.
Because the function has a discontinuity at x=2x = 2, it changes sign across the asymptote without ever touching y=0y = 0, producing a false positive.
Can a function have a perfectly valid root that leaves no sign change behind?

False Negatives: Even-Multiplicity Roots

The multiplicity of a root is the number of times its factor (x−r)(x - r) appears in the factored form of the polynomial equation.