Is this property true? Suppose $f$ is a continuous function on the compact set $[0,1]$. Is true that given $\epsilon>0$, there exist $\delta>0$ such that for any partition $P=\{x=x_0<x_1,...,<x_{i-1}<y=x_i\}$ of the interval $[x,y]$, with $|x-y|<\delta$, we have that $$\sum_{n=1}^i|f(x_n)-f(x_{n-1})|<\epsilon$$

Consider the function
$$f(x)=\begin{cases}
0,&\text{if }x=0\\
x\sin\frac1x,&\text{if }0<x\le 1\;;
\end{cases}$$
clearly $f$ is continuous, but you can make
$$\sum_{n=1}^i|f(x_n)-f(x_{n-1})|$$
as large as you like when $x=0$, even if $y$ is very small: if you choose the right partition, you’re essentially adding up terms of the harmonic series.
