This line is drawn in plan first and I turned the POINTSON. After this, I pulled the points to different height.
However, when I offset this three dimensional line, the new line doesn’t actually joined.
I will really appreciate if anyone can help me. 
Thanks
AL
Hi Alex- Offset is a 2d operation, (The 3d offset of a curve is a tube or pipe- more like a sueface) normally the plane of offset is the plane of the curve itself- Once you have a 3d line, all bets are off- if you extend the segments of your offset, most likely most or all of them will not meet.
You can try BlendCrv to make up the gaps.
-Pascal