What is the meaning of Dashing[Block]? What is the 12 in Directive[Black, 12]? Did you mean to specify the Thickness? That's not the way to do it. Your function l[r] behaves very differently with exact rational values of r and with floating-point values. Compare the time it takes for l[.5] and for l[1/2]. Why do you resort to ListLinePlot with rational values of r and not just do this:

Plot[l[r], {r, .01, .8}, PlotStyle -> Black, PlotRange -> {0, 0.5}, 
 Frame -> True, RotateLabel -> False, 
 FrameStyle -> Directive[Black, Thickness[.004]], 
 FrameLabel -> {Style["r", 13, Bold, Black], 
   Style["N", 13, Bold, Black]}, 
 PlotLabel -> Style["b", 13, Bold, Black]]

What version are you running? Under version 13.2 your plot is continuous and looks fine.

You do notice that you have PlotStyle->{Dashing[Block],Black} Do you really want a dashed plot when you are asking why you are seeing gaps in your plot, but I don't think that is your issue?

What happens if you include PlotPoints->100 or PlotPoints->250 to increase the number of sampling points? Pushing that up too much will slow it down, but may improve the quality.