TITLE: Singular chains in Spivak's Calculus on Manifolds
QUESTION [9 upvotes]: On page 90 of Calculus on Manifolds, Spivak defines the
pull-back $f^*\omega$ of a differential form by a differentiable 
map by the usual formula.  On page 97, he defines a singular
$k$-cube as a continuous map $c:[0,1]^k\to\mathbb{R}^n$.  Finally,
on page 101, he defines the integral of a differential $k$-form
against a singular $k$-cube by the formula 
$\int_c\omega=\int_{[0,1]^k} c^*\omega$.   
I don't see, however,
where he defines the pull back $c^*\omega$ of a differentiable form by
a continuous map (or how he could).
So my questions are: (a) Is this an error (or am I missing
something)? (b) Have other people noticed it?

REPLY [2 votes]: Continuity is not enough. However   Lipschitz continuity would  do. Lipschitz continuous maps are differentiable  almost everywhere  and  you can define   the pullback almost everywhere. For more on this point of view,    have a look at a book on  geometric measure theory, e.g.,  Frank Morgan's: Geometric Measure Theory. A Beginner's Guide.
Another beautiful reference is   the classic Geometric Integration Theory by H. Whitney (now available in Dover)