I'm reading Complex Analysis by Ahlfors and some sentences don't make sense to me.
- "The arc is differentiable if $z'(t)$ exists and is continuous (the term continuously differentiable is too unwieldy)" (p. 68).
The first question I have is: is an arc differentiable if it has a discontinuous derivative? I think a function is usually differentiable if it has a derivative and continuity does not really matter.
Next, I do not understand what the author means by "the term continuously differentiable is too unwieldy."
2."An arc is piecewise differentiable or piecewise regular if the same conditions hold except for a finite number of values $t$; at these points $z(t)$ shall still be continuous with left and right derivatives which are equal to the left and right limits of $z'(t)$ and, in the case of a piecewise regular arc, $\neq0$." (p. 68).
I don't quite understand the second half (from "at these points$\ldots$"). Could you break down this sentence into $2$ or $3$ sentences so that it's easier to understand?