Would a triangle wave have finite or infinite sinusoidal components?
a triangle wave is continuous


Quote from here: -


  The triangle wave has no discontinuous jumps, but the slope changes
  discontinuously twice per cycle


Having the slope change discontinuously also means an infinite range of sinusoidal components.

For instance, if you time integrated a square wave you produce a triangle wave but, all the hamonics of the original square wave are still present after the time integration: -