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I'm reading Complex analysis by Alfors and some sentences don't make much sense to me.

  1. "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?

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1 Answer 1

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1) He's defining "differentiable arc" to mean what is more properly called "continuously differentiable". He doesn't want to call it a "continuously differentiable arc" because that phrase is too unwieldy, i.e. long and awkward; he's probably going to use this a lot, and doesn't want to waste his ink and the reader's time.

2) There may be a finite number of points $t_i$ at which the arc $z(t_i)$ is not differentiable. However, $z$ is still continuous at $t_i$, the left derivative $D_{-} z(t_i) = \lim_{t \to t_i-} \dfrac{z(t) - z(t_i)}{t - t_i}$ and the right derivative $D_{+} z(t_i) = \lim_{t \to t_i+} \dfrac{z(t) - z(t_i)}{t - t_i}$ are required to exist, $D_{-}(t_i) = \lim_{t \to t_i-} z'(t)$ and $D_{+}(t_i) = \lim_{t \to t_i+} z'(t)$. Moreover, if it's supposed to be piecewise regular, $D_{-}(t_i)$ and $D_{+}(t_i)$ are nonzero.

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Thanks for reply! You answered my questions thoroughly! –  Tengu Nov 21 '12 at 2:15

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