Removing vertical lines due to sudden jumps in gnuplot

Question

I am trying to plot a function that contains discontinuities in gnuplot. As a result, gnuplot automatically draws a vertical line connecting the jump discontinuities. I would like to remove this line. I have looked around and found two solutions, none of which worked: One solution was to use smooth unique when plotting, and the other one was to define the function in a conditional form and remove the discontinuity manually. The first solution simply did not make any changes to the plot (at least visually). The second solution seemed to move the location of the jump discontinuity to left or right, not get rid of the vertical line. Please note that I would like to plot with lines. I know with points works, but I do not wish to plot with points.

set sample 10000
N=50
l1(x)=2*cosh(1/x)
l2(x)=2*sinh(1/x)
Z(x)=l1(x)**N+l2(x)**N
e(x)=(-1/Z(x))*(l2(x)*l1(x)**(N-1)+l1(x)*l2(x)**(N-1))
plot e(x)

Produces:

Answer 1

If all you need to do is to remove the vertical line at the singularity you could use conditional plotting:

plot (x<0 ? 1/x : 1/0)  w l ls 1, (x>0 ? 1/x : 1/0) w l ls 1

However, your function is more complicated: it cannot be numerically evaluated in a region around 0:

set grid
set xrange [-0.3:0.3]
plot e(x) with linespoints

If Mathematica is to be trusted, the function e(x) goes to 1 and -1 as x approaches 0 from the left and the right, respectively. However, you see in the picture above that gnuplot fails to properly evaluate the function already at x=0.1. print e(0.1) gives -0.0, and print e(0.05) already gives NaN. In this region the numerator and denominator of the function e(x) get too large to be handled with floating point numbers.

You can either exclude this region using conditional plotting,

plot (x<-0.15 ? e(x) : 1/0)  w l ls 1, (x>0.15 ? e(x) : 1/0) w l ls 1

or you have to rewrite the function e(x) so you avoid extremely large values in its evaluation (if that is possible). Alternatively you can use a software package that can switch to higher precision, such as Mathematica.