Let $\gamma$ be a piecewise smooth curve in $\mathbb{R}^n$. Recall that the *centroid* of $\gamma$ is the point $(\overline{x}, \overline{y})$ where $\overline{x}$ is the average value of $x$ on $\gamma$ and $\overline{y}$ is the average value of $y$ on $\gamma$:

$$\overline{x} = \frac{1}{\text{Length}(\gamma)} \int_\gamma x\, d\gamma, \hspace{1cm} \overline{y} = \frac{1}{\text{Length}(\gamma)} \int_\gamma y\, d\gamma$$

My question is: if $\gamma_n$ is a sequence of piecewise smooth curves which converge uniformly to a piecewise smooth curve $\gamma$, is it true that $(\overline{x_n}, \overline{y_n}) \to (\overline{x}, \overline{y})$? If it is more convenient to replace "piecewise smooth" with "rectifiable" or something else, I don't mind.

A hint that this might not be completely trivial is the observation that $\text{Length}(\gamma_n)$ need not converge to $\text{Length}(\gamma)$: the standard example is a sequence of finer and finer staircase curves converging uniformly to a diagonal line. However, the sequence of centroids does converge to the right limit in this example.