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Is the curve defined by:

γ(t) = (t,t) for 0≤t≤1 and (2-t,2-t) for 1≤t≤2 piecewise smooth?

My logic says yes because one can break it into a finite number of smooth curves (two in this case), but something doesn't add up for me.

Yarduza
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    It indeed has two smooth pieces, that's it. –  Aug 09 '15 at 13:12
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    Yes. This is just $(1-|1-t|,1-|1-t|), 0\leq t\leq 1$. This curve is smooth except at $t=1$, so it is piecewise smooth. – MPW Aug 09 '15 at 13:15
  • Thank you, my main problem was that the lines that it defined were overlapping, I thought it might alter the result. – Yarduza Aug 09 '15 at 13:17
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    Well, remember to check continuity, but I think it's ok. –  Aug 09 '15 at 13:18

1 Answers1

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It is piece-wise smooth, but if it does not seem meaningful that it overlaps itself, you may want to know that it is called "self-intersecting". Often we want to allow self-intersecting curves so that we can easily handle contour integrals on piecewise smooth curves by being able to add them together simply by joining the ends.

user21820
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