The technique used below appears frequently in Complex Analysis but I have difficulty formally proving the result. So we have a continuous function $f$ on some set including a contour as in $C^+$ below, and we approximate the integral of $\int_{C^+} f$ as the limit of $\int_{C_\epsilon^+}f$. All it says is that an easy continuity argument shows it. So essentially we need to prove that $$\vert\int_{C^+}f - \int_{C_\epsilon^+}f\vert<\epsilon$$ for any small $\epsilon>0$. The difference of the integrals would be a contour integral around another rectangle which is the difference between the two rectangles $C^+$ and $C_\epsilon^+$. I thought about using the maximal inequality of the contour integral, however, since the diameter of the rectangle would be $2(\epsilon + d)$, where $d$ is the length of the rectangle, it does not go to $0$ as $\epsilon \to 0$. How can I prove this easy continuity argument?
