Covering the graph of a function

Question

$f:[0,1]→\mathbb{R}, ∣f(x)−f(y)∣≤∣x−y∣ \forall x,y∈[0,1] $.

Show that the graph of $f$ can be covered by rectangles that the sum of their shorter sides is less than $ε$, $ \forall ε>0$.

I am not sure how to conclude this. Of course the constant and linear functions can be covered like this, but I could not come up with a soultion for other functions having this property.

EDIT: I think one has to show that there is a partition on which in every interval $f$ can be approximated by different linear functions.

Answer 1

Let partition of $[x,y]$ such that:

$P=\lbrace t_0,t_1,\ldots,t_{n-1},t_n\rbrace$, such that length $\mathcal{l}([x,y])=\frac{y-x}{n}$, now;

$\sum_{i=1}^{n}|f(t_i)-f(t_{i-1})|\leq\sum_{i=1}^{n}|t_{i-1}-t_{i}|=\frac{|x-y|}{n}$ by archimedian property you have that

$\sum_{i=0}^{n}|f(t_i)-f(t_{i-1})|\leq \varepsilon|x-y|$