information:classification, structural:scholar_edition
Motivation

This captures an intuitive property of continuous functions over the real numbers: given f {\displaystyle f} continuous on {\displaystyle } with the known values f (1) = 3 {\displaystyle f(1)=3} and f (2) = 5 {\displaystyle f(2)=5}, then the graph of y = f (x) {\displaystyle y=f(x)} must pass through the horizontal line y = 4 {\displaystyle y=4} while x {\displaystyle x} moves from 1 {\displaystyle 1} to 2 {\displaystyle 2}. It represents the idea that the graph of a continuous function on a closed interval can be drawn without lifting a pencil from the paper.