I'm reading Complex analysis by Alfors and some sentences don't make much 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 that: Is an arc differentiable if it has uncontinuous 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 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, $\neq$0."(P. 68)
I don't quite understand the second half (From ";at these points..."). Could you break down this sentence into 2 or 3 sentences so that it's easier to understand?