Piecewise linear homotopy

Definition
Suppose $$f,g:X \to Y$$ are continuous maps with $$Y$$ a subset of a (possibly infinite-dimensional) Euclidean space with the subspace topology. Suppose there exist continuous maps $$f_0,f_1,f_2,\dots,f_n:X \to Y$$ such that $$f = f_0$$ and $$g = f_n$$ and linear homotopies $$F_{i(i+1)}$$ between each $$f_i$$ and $$f_{i+1}$$. Then, we can define a composite of homotopies $$F_{i(i+1)}$$ which is a homotopy from $$f$$ to $$g$$. A homotopy obtained in this way is termed a piecewise linear homotopy.