Recall the proof of Lemma 2.2.8 in which a Lebesgue number was shown to exist. This is a number
which satisfies the conclusion of the following lemma whose proof is contained in the earlier
lemma.

Lemma 4.2.6Let C denote a set of open intervals covering

[a,b]

. Then there exists δ > 0 suchthat any interval of length no more than 2δ is contained in one of the intervals of C.

One can also find the length of a C^{1} curve. Letting γ :

[a,b]

→ ℝ^{p} be C^{1}, an approximation to the
length or total variation of the curve would be

∑n
|γ (ti)− γ(ti−1)|
k=1

and the length is defined above to be the least upper bound of all such sums. By differentiability,

′
γ (t)− γ (s) = γ (s)(t− s)+ o(t− s)

Then by continuity of the derivative, for each s ∈

[a,b]

, there is δ_{s}> 0 such that if x,y ∈

(s − δs,s+ δs)

,
then

|o (y− x)|

<

-ε-
b−a

|y− x|

. From Lemma 4.2.6 above, there exists δ > 0 such that any interval of length
less than δ is contained on one of the intervals