Let $X$ be a Banach space and assume that $a. For a function $f\colon\left [a,b\right]\to X$ we define a generalization of Riemann integral as follows: a point $u\in X$ we call the integral of $f$ if for any $\epsilon>0$ there is a $\delta>0$ such that for $a=t_0
One can show that contiuous functions are integrable in the sense of the definition above.
My question is how to determine the integral of the function $f:\left[0,1\right]\rightarrow c_0$ given by $f(x)=\frac{1}{2}\left(x-\frac{1}{2^n}\right)e_n+\frac{1}{4}\left(\frac{1}{2^{n-1}}-x\right)e_{n+1}$, for $x\in\left[\frac{1}{2^n}, \frac{1}{2^{n-1}}\right]$, $n\in\mathbb{N}$ and $f(0)=0$, where $e_n=(0,...,0,1,0,...,0)$ with $1$ being a the $n$-th place?
Notice that $f$ is continuous so it is integrable.
I guess that the integral may be of the form $\sum_{n=1}^{\infty} \frac{1}{2^{n+1}}e_n$ or something like this but can not deal with it, so I will be very grateful for your help.