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4.9: The Intermediate Value Property
https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/04%3A_Function_Limits_and_Continuity/4.09%3A_The_Intermediate_Value_Property
Geometrically, if $A \subseteq E ^ { 1 } ,$ this means that the curve $y = f ( x )$ meets all horizontal lines $y = q ,$ for $q$ between $f ( p )$ and \( f \left( p _ { 1 } \right)...Geometrically, if $A \subseteq E ^ { 1 } ,$ this means that the curve $y = f ( x )$ meets all horizontal lines $y = q ,$ for $q$ between $f ( p )$ and $f \left( p _ { 1 } \right) .$ For example, in Figure 13 in $§ 1 ,$ we have a "smooth" curve that cuts each horizontal line $y = q$ between $f ( 0 )$ and $f \left( p _ { 1 } \right) ;$ so $f$ has the Darboux property on $\left[ 0 , p _ { 1 } \right] .$ In Figures 14 and $15 ,$ there is a "gap" a…

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