Take a curve $z(t) = x(t) + i y(t), a \leq t \leq b$ in the complex plane.
Consider a partition $Q= (a = t_0, t_1, \dots, t_n = b)$ of $[a,b]$.
Associate a partition $P = (z(t_0), \dots, z(t_n))$ and define its norm as
$$N_P := \max_k \{|z_k-z_{k-1}|\}$$
Suppose $f$ is a complex-valued function defined on the curve in the complex plane. Choose $t_k^* \in [t_{k-1}, t_k]$ and define
$$S_P = \sum_{k=1}^n f(z(t_k^*))(z_k-z_{k-1})$$
My book then gives the following definition:
If $\lim_{N_P \to 0} S_P$ exists and is independent of the chosen points $t_k^*$, we write $$\int_{z=z(t)} f(z) dz$$ for the limit.
Question:
Does this mean, denoting the limit with $L$:
For all $\epsilon > 0$, there is some $\delta > 0$ such that if $P$ is a partition as above with $N_P < \delta$ and if $t_k^*$ is arbitrarily chosen as above, then
$$|S_P - L| < \epsilon$$
Yes, that's the meaning of integral of $f$ along the curve $z$.