If $\gamma:[a,b]\rightarrow \mathbb{C}$ is a piecewise smooth path and if $\beta:[c,d]\rightarrow \mathbb{C}$ is a path that can be obtained from $\gamma$ by making a piecewise smooth change of parameter, then $\beta$ is also piecewise smooth.
We know that $\beta=\gamma h$ where $h$ is a piecewise smooth path and so we need $\beta(t)=\gamma(h(t))$ to make sense and for this we could not take the same partition that gives me the fact that $h$ is piecewise smooth and this would serve me to prove that $\beta$ is piecewise smooth path? Thank you in advance