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Formally, the definition can be stated as follows. Let <math>G</math> be a subset of the [[Euclidean space]] <math>\R^n</math> and let
<math display="block">\varphi \colon G \to \R \cup \{ - \infty \}</math>
be an [[semi-continuity|upper semi-continuous function]]. Then, <math>\varphi </math> is called ''subharmonic'' if for any [[closed ball]] <math>\overline{B(x,r)}</math> of center <math>x</math> and radius <math>r</math> contained in <math>G</math> and every [[real number|real]]-valued [[continuous function]] <math>h</math> on <math>\overline{B(x,r)}</math> that is [[harmonic function|harmonic]] in <math>B(x,r)</math> and satisfies <math>\varphi(y) \leq h(y)</math> for all <math>y</math> on the [[boundary (topology)|boundary]] <math>\partial B(x,r)</math> of <math>B(x,r)</math>, we have <math>\varphi(y) \leq h(y)</math> for all <math>y \in B(x,r).</math>
Note that by the above, the function which is identically −∞ is subharmonic, but some authors exclude this function by definition.
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