finitely linear

A function $f\: I \to Y$, where I is a closed interval of the real line, is called finitely linear provided there exists a positive integer m such that I can be decomposed, for each $\varepsilon > 0$, into a finite number of closed subintervals $I_1, I_2, \cdots, I_k$ each of length less than $\varepsilon$ and with the property that the set f(I_i) meets at most m of the sets $f(I_1), f(I_2), \cdots, f(I_k)$ for $i = 1, 2, \cdots, k$.