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A simple proof is as follows: if there were a point <var>x</var><sub>0</sub> within <var>Ω</var> for which {{math|<var>F</var>(<var>x</var><sub>0</sub>) ≠ 0}}, then the continuity of {{math|<var>F</var>}} would require the existence of a [[neighborhood (mathematics)|neighborhood]] of <var>x</var><sub>0</sub> in which the value of {{math|<var>F</var>}} was nonzero, and in particular of the same sign than in <var>x</var><sub>0</sub>. Since such a neighborhood <var>N</var>, which can be taken to be arbitrarily small, must however be of a nonzero width on the real line, the integral of {{math|<var>F</var>}} over <var>N</var> would evaluate to a nonzero value. However, since <var>x</var><sub>0</sub> is part of the ''open'' set <var>Ω</var>, all neighborhoods of <var>x</var><sub>0</sub> smaller than the distance of <var>x</var><sub>0</sub> to the frontier of <var>Ω</var> are included within it, and so the integral of {{math|<var>F</var>}} over them must evaluate to zero. Having reached the contradiction that {{math|∫<sub><var>N</var></sub><var>F</var>(<var>x</var>) <var>dx</var>}} must be both zero and nonzero, the initial hypothesis must be wrong, and thus there is no <var>x</var><sub>0</sub> in <var>Ω</var> for which {{math|<var>F</var>(<var>x</var><sub>0</sub>) ≠ 0}}.
The theorem is easily generalized to [[multivariate calculus|multivariate function]]s, replacing intervals with the more general concept of connected [[open set]]s, that is, [[Domain (
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