In Cauchy's integral theorem (see: https://en.wikipedia.org/wiki/Cauchy%27s_integral_theorem), does the $\gamma$ representation of the $\Gamma\subset\mathbb{C}$ curve always has to be smooth (i.e. at least differentiable)?
As I understand, the complex curve integral on a $\Gamma$ curve is defined as a kind of Stieltjes-integral:
$$\int_{\Gamma}f\left(z\right)dz\dot{=}\int_{a}^{b}f\left(\gamma\left(t\right)\right)d\gamma\left(t\right),$$
where $\gamma:\left[a,b\right]\rightarrow\Gamma$. If $\gamma$ is differentiable (except at some countable points), then
$$\int_{\Gamma}f\left(z\right)dz=\int_{a}^{b}f\left(\gamma\left(t\right)\right)\gamma'\left(t\right)dt.$$
So I would say $\gamma$ (and as a result $\Gamma$) doesn't have to be smooth to define the $\int_{\Gamma}f\left(z\right)dz$ complex integral. (But I think $\gamma$ at least should be a function with bounded variation, because in this case the Stieltjes integral is reasonable. E.g. $\Gamma$ shouldn't be a curve like a typical trajectory of a Wiener-process...I mean in this case the integral doesn't always exist...)
However, I'm not sure if this is the case at Cauchy's integral theorem. In Cauchy's integral theorem does $\gamma$ (and $\Gamma$) have to be smooth? I saw a proof where it was used that $\gamma'$ had to exist.
In short: what properties $\gamma$ (and $\Gamma$) has to have in order to have a well defined definition for the complex curve integral, and what should hold in order to use Cauchy's integral theorem?
I've just started to learn complex analysis and I was wondering on these questions...