A fineliner (one word) is a pen with a felt tip, almost like a felt tip marker but smoother and more precise. in the case of Montblanc pens, they are also spring loaded to prevent damage to the nip.
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It's kind of like a texter but much thinner and is usually black, although you can get ones that are different colours. They're used for outlining images to make it stand out or make the lines thicker/bolder.
Fine line, fine sign, design line, smart art
You might try a fine tip paint pen.
Broad line markers have a thicker, typically flat edge which will produce a thick line on a paper. Fine line markers will produce a thin line and typically come to a thin point rather than an edge. Also, fine line markers are meant for drawing outlines, while broad line markers are meant for filling/coloring in those outlines.
I like the Sharpie and Pilot brands but these days there are a lot of options. India ink with a brush is a nice effect, but a fine point or extra fine point rolling ball pen allows for more control.
A continuous line means a single-unbroken line. In the case of visual art it is when an entire picture is made without lifting the tool (pen, pencil etc) from the surface (paper, canvas etc).