/ 1   if x > a
step(x) = -|
           \ 0   else

In Gnuplot this can be achieved by using the ternary operator:

step(x) = x>a ? 1 : 0

Which is a simple if-else statement and means step(x)=1 if x>a else step(x)=0.

If we plot this function we get Fig. 1.

Fig. 1 Continuous plot of the step function (code to produce this figure)

As you can see this will result in a continuous plot. If we want a discontinuity in the plot, we have to create two separate functions that are only piecewise defined. This can be achieved by using 1/0 that will result in a undefined value.

f(x) = x<a  ? 1 : 1/0
g(x) = x>=a ? 1 : 1/0 

Plotting both functions will result in Fig. 2.

Fig. 2 Discontinuous plot of the step function (code to produce this figure)

The ternary operator can also be used in an iterative way. For example if we want to define a rectangular function that is given by

           / 0     if abs(x) > 0.5
rect(x) = -| 0.5   if abs(x) = 0.5
           \ 1     if abs(x) < 0.5

we can use the following statement in Gnuplot to define it:

rect(x) = abs(x)>0.5 ? 0 : abs(x)<0.5 ? 1 : 0.5

In Fig. 3 you can see a plot of this function. To produce the sharp edges of the rectangular function we use a higher number of sampling points (also in Fig. 1 for the step function). In order to plot a function Gnuplot calculates 100 points of the given function and draw a line through them. This can be set to another value with the set samples <value> command.

Fig. 3 Plot of the above defined rect(x) function (code to produce this figure)

Fig. 1 Plot of the data from plotting_data1.dat with non-coninuous lines between its points (code to produce this figure)

In Gnuplot exists no line style that can do this directly. But with a little trick it is very easy to achieve. Since Gnuplot 4.4. there exists the property pointinterval (see the documentation) in combination with the plotting style linespoints. This property plots not every single point, but only every second for a value of 2 and so on. But if we use the value -1 it tells Gnuplot to insert a little gap between the points and the line. The size of the gap can be set by the pointintervalbox property.

set style line 1 lc rgb '#0060ad' lt 1 lw 2 pt 7 pi -1 ps 1.5
set pointintervalbox 3

We specify a point interval pi of -1 and a point size of 1.5, in addition we set the the gap to a point size of 3. Now we can plot our data with the linespoints style.

plot 'plotting_data1.dat' with linespoints ls 1

Using the same data as in the first plot of the gnuplot basics tutorial Plotting data we will get Fig. 1 as a result.