Definition of a piecewise smooth path

Question

In Apostol's "Mathematical Analysis" $($page 435$)$, a piecewise smooth path in the complex plane, say $f$, is defined as a path in the complex plane that has bounded derivative $f'$ which is continuous everywhere except possibly at a finite number of points, and at these exceptional points it is required that both right- and left-hand derivatives exist. Specifically, if $f:[a,b]\rightarrow C$ is a complex-valued function on the compact interval $[a,b]$, then $f$ is said to be piecewise smooth if there exists a partition $\{x_0,x_1,...x_n\}$ of $[a,b]$ such that $f$ has a bounded and continuous derivative on each open subinterval $(x_{i-1},x_i)$ and has one-sided derivatives at the endpoints $x_{i-1}$ and $x_i$.

However, in the definition of a piecewise smooth path in most books on complex analysis, it is required that the restriction of $f$ to each compact subinterval $[x_{i-1},x_i]$ has a "continuous" derivative on $[x_{i-1},x_i]$. The requirement in this definition is stronger than that of Apostol since at the endpoints $x_{i-1}$ and $x_i$ it requires that $\lim_{x\rightarrow x_{i-1}+}f'(x)=f'_{+}(x_{i-1})$ and $\lim_{x\rightarrow x_i-}f'(x)=f'_{-}(x_i)$. I would like to know whether Apostol's definition is equivalent to this definition. Thanks for your feedback.

Answer 1

The two definitions are not equivalent.

For example, take $a = -1,$ $b = 1,$ $(x_0, x_1, x_2) = (-1, 0, 1),$ and: $$ \gamma \colon [-1, 1] \to \mathbb{C}, \ x \mapsto \begin{cases} 0 & \text{if } x = 0, \\ x + ix^2\sin\left(\frac1x\right) & \text{if } x \ne 0. \end{cases} $$ The path $\gamma$ has a bounded derivative everywhere: $$ \gamma'(x) = \begin{cases} 1 & \text{if } x = 0, \\ 1 + 2ix\sin\left(\frac1x\right) - i\cos\left(\frac1x\right) & \text{if } x \ne 0. \end{cases} $$

According to Apostol's definition, $\gamma$ is piecewise smooth, because $\gamma'$ is continuous on $[x_0, x_1)$ and $(x_1, x_2],$ and the one-sided derivatives $\gamma'_-(x_1)$ and $\gamma'_+(x_1)$ both exist. (Both equal $\gamma'(0) = 1.$)

I'm not aware of another book that uses the same definition as Apostol. I'll update this answer if I find any.

According to Ahlfors, Complex Analysis (3rd ed. 1979), p.68 (and every other book I've looked at), $\gamma$ is not piecewise smooth, because the one-sided limits $\lim_{x\to x_1-}\gamma'(x)$ and $\lim_{x\to x_1+}\gamma'(x)$ do not exist.