Why doesn't exist a Cousin's lemma for left-tagged partitions? I am thinking of the possible validity of a
 statement like this:
given any positive mapping $\delta$ on $[a,b]$, there exists a partition $\, a=a_0<a_1<\cdots<a_n=b \,$ of $ \,[a,b] \,$ so that $$a_r - a_{r-1} < \delta (a_{r-1})\qquad(r=1,\ldots,n)$$ Jean Mawhin says it is not true at the end of his contribution in Fulvia Skof (Ed.) Giuseppe Peano between Mathematics and Logic (2011).
I cannot understand if the reason is banal or not, so I need your help to know why it is not true. Thank you in advance.
(the question is tied to the need of the so-called straddle lemma in the introduction of the Henstock-Kurzweil integral)

A counterexample should illustrate.  Consider the interval $[0,2]$
define your function $\delta:[0,2]\to R^+$ by requiring that
$\delta(x)< 1-x$ for all $x\in [0,1]$ and any other values you wish elsewhere.
If an interval $[a_{r-1},a_r]$ in the first half of the interval satisfies $a_r-a_{r-1}<\delta(a_{r-1})$ then
$a_r< a_{r-1} + \delta(a_{r-1})<1$.  So you never reach the point $1$ in your
partition.  
But that does not mean that there is no left-tagged partitions possible if
you make suitable adjustments.  John Hagood came up with a very nice idea
for this and we used it to integrate Dini derivatives of continuous functions.
If you care to pursue see

MR2202919 (2006i:26010)  Hagood, John W. ;  Thomson, Brian S.
Recovering a function from a Dini derivative.   Amer. Math. Monthly  113  (2006),  no. 1, 34--46.   Download link here.

