Is the existence of a continuous top form sufficient for orientability of a smooth manifold?

Question

I'm currently working through Tu's Introduction to Manifolds. He first defines orientability in terms of the existence of orientations on the tangent spaces which can be locally represented by a continuous frame, and then goes on to show that this is equivalent with the existence of a smooth top form which vanishes nowhere on the manifold. My question is: why do we need this form to be smooth? Wouldn't continuity already suffice? I can't figure out the motivation from the proof he gives.