Continuous Coloring

Geometry Level 5

Let \(T\) be an equilateral triangle of side length \(1\), whose vertices are located at \((0,0), (1,0) \) and \(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\). Let \((f_i), i=1,2,3, \ldots \) be continuous functions such that, \[f_i : [0,1] \to T\] i.e., each point in \([0,1]\) is mapped to some point in the triangle \(T\) by each function \(f_i\).

Think of each \(f_i\) as a pen with color \(i\). So that each \(f_i\) colors its range (which is some portion of the triangle \(T\) ) with the color \(i\). E.g., if \(f_1(1/3)=(\frac{1}{2},\frac{1}{2})\), this means that the point \((\frac{1}{2},\frac{1}{2})\) is colored '1' by the function \(f_1\).

Find the minimum number of continuous functions \(f_i\) needed to color the entire triangle.

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