Continuous Coloring

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.