The space you’re talking about is usually constructed by taking the closure of smooth compactly supported functions with respect to the Sobolev norm. In 1 dimension, every W^m,p function admits a continuous representative*, which you can require to vanish at the boundary.

*This is because W^1,1 in 1 dimension consists of precisely the absolutely continuous functions, and W^1,p consists of Holder continuous functions by Morrey’s embedding. For higher Sobolev spaces the situation only improves.