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<math>\{p\in M:\varphi(p)\le y\}</math> that contain at least one point at which the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>x\ </math>
<ref name="dAFrLa06">Michele d'Amico, Patrizio Frosini, Claudia Landi, ''Using matching distance in Size Theory: a survey'', International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.</ref>. The concept of size function can be easily extended to the case of a measuring function <math>\varphi:M\to \mathbb{R}^k</math>, where <math>\mathbb{R}^k</math> is endowed with the usual partial order
<ref name="BiCeFr08">Silvia Biasotti, Andrea Cerri, Patrizio Frosini, Claudia Landi, ''Multidimensional size functions for shape comparison'', Journal of Mathematical Imaging and Vision 32:161-179, 2008.</ref>.
[[Image:SFesWiki.PNG|1065px|thumb|left|''An example of size function. (A) A size pair <math>(M,\varphi:M\to\mathbb{R})</math>, where <math>M</math> is the blue curve and <math>\varphi:M\to \mathbb{R}</math> is the height function. (B) The set <math>\{p\in M:\varphi(p)\le b\}</math> is depicted in green. (C) The set of points at which the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>a</math> is depicted in red. (D) Two connected component of the set <math>\{p\in M:\varphi(p)\le b\}</math> contain at least one point at which the [[measuring function]] <math>\varphi</math> takes a value smaller than or equal to <math>a</math>. (E) The value of the size function <math>\ell_{(M,\varphi)}</math> in the point <math>(a,b)</math> is equal to <math>2</math>.'']]
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