Theorem 5.2.1Let
be a path of length
and
be a function, continuous at
every point of
. Moreover suppose that there exists a positive real
number
such that
. Then:

The proof is based on the fact that

which can be proven using Riemann sums.

Example 5.2.2
Let
be the upper half of the unit-circle; then
. Consider
the function
; we have:
, therefore
. It follows that
.