The first example of a homotopy that a topology student is introduced to is a homotopy between functions. A homotopy between two functions <nowiki>"f" and "g" provides a collection of functions that start at "f" and eventually become "g". What this means is that this homotopy takes a parameter and gives a function, and there is a possible parameter that gives "f" and one that gives "g"</nowiki>. Typically, we choose the interval <nowiki>"</nowiki>[0, 1]<nowiki>"</nowiki> because this has a starting and ending point, and numbers are allowed to vary smoothly within <nowiki>"</nowiki>[0, 1]<nowiki>"</nowiki>. To output a function, this means that the homotopy takes a second parameter, which is the argument of the output function. That is, if <nowiki>"</nowiki>f<nowiki>"</nowiki> takes a number in the interval and outputs pairs of real numbers, the homotopy takes a number in the interval as the parameter and another number in the interval for <nowiki>"</nowiki>f<nowiki>"</nowiki>. The homotopy then outputs the interpolated function evaluated at the given pair. That is, if <nowiki>"</nowiki>H<nowiki>"</nowiki> is a homotopy from <nowiki>"</nowiki>f<nowiki>"</nowiki> to <nowiki>"</nowiki>g<nowiki>"</nowiki>, then we know formula_1where formula_2and formula_3. 