We want piecewise continuity for any contours in complex analysis. What does this refer to?
I imagine it refers to the nature of referring to each line arc being continuous. E.g. We want continuity on open neighborhoods of each segment. So we only need continuity on each of the curves here, not including their endpoints:
Actually you need continuity everywhere, not just piecewise.
The sharp bends in your drawings are not points of discontinuity -- the curve is perfectly continuous there, but it may (depending on the parameterization) fail to be differentiable there.
What your textbook probably requires is that the curve must be a piecewise differentiable -- that is, a concatenation of differentiable curves on closed real intervals, where each piece is differentiable at the endpoints too, even if its derivative doesn't match that of its neighboring piece. This condition is actually slightly stronger than what is strictly needed (I think "rectifiable" will do) but it is easy to state, easy to understand, still sufficient for many applications, and relatively simple to prove sufficient.
What can go wrong if the curve is too "wild" -- for example if part of the curve goes like $ t\mapsto (t,t\sin(t^{-2})) $ on $(0,1)$ -- is that the contour integrals may reduce to real integrals that fail to converge as Riemann integrals at all.