Piecewise Smooth Curve

Question

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.

Answer 1

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.