I am going through J.W. Anderson's book $Hyperbolic$ $Geometry$, and I come along to the following statement made: "We note here that the concatenation of piecewise differentiable paths is again piecewise differentiable, while the concatenation of differentiable paths is not necessarily differentiable".
For the second part about differentiable paths, I understand it as I just need to take the paths $\overrightarrow{r}(x(t),y(t))$, $x(t) = t$, $y(t) = t$ and $\overrightarrow{r_1}(x_1(t),y_1(t))$, $x_1(t) = -t$, $y_1(t) = t$, $t\geq 0$ for both. Then there is a cusp when $t=0$ so the concatenation of the two paths is not differentiable. However, how can I prove the first statement that the concatenation of piecewise differentiable paths is again piecewise differentiable?
Ben