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I have a rather simple question. I have the curve

##

C(t) =

\begin{cases}

1 + it & \text{if}~ 0 \le t \le 2 \\

(t-1) + 2i & \text{if }~ 2 \le t \le 3

\end{cases}

##

which is obviously formed from the two curves. This curve is regarded as an arc if the functions ##x(t)## and ##y(t)## are continuous real mappings. Clearly each individual curve has continuous real mappings. But here is my concern: what if the two curves did not coincide somewhere in the complex plane, that is, they were not joined somewhere? Would the curve ##C(t)## no longer be an arc?

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# Continuity of an arc in the complex plane

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