For the proof of Fundamental Lemma 3.1 on the page 400 of K.T. Chen's 1957 paper Integration of paths--A faithful representation of paths by noncommutative formal power series, it requires the path $\beta(t)$ to be regular. However, I was wondering if I replace the regular path $\beta(t)$ with a continuous injective path with finite number of irregularities, would the arguments in the proof still work? Because the construction of the cube is based on the property of continuity and injectivity of the path, and one can still find such a cube for an irregular path as long as it is continuous. The only problem is the evaluation of the integral $$\int g(x_1(\beta_i(t)))....\frac{dx_1(\beta_i(t))}{dt}dt$$ as the derivative $\frac{dx_1(\beta_i(t))}{dt}$ might not be continuous. However, if we require the path have only finite irregularities, the discontinuity of the derivative will be only finitely many and one will still be able to evaluate the integral.
So the question summarized:
- How would irregularities affect the construction of the cube in the proof?
- Can the arguments in the proof be fixed such that the proof works for an irregular path?
