Let's consider plotting
sin(1/x) -1<=x<=1.Near x= 0, a naive plotting program based on sampling at fixed intervals (say every 0.001) will give us a plot which is effectively a collection of random numbers between -1 and 1. This is a poor (unfaithful) description of the function.
Consider plotting the Intervals sin(1/[x,x+0.001]) instead. This means drawing the rectangles of appropriate height (and width 0.001) that enclose the points. This looks like a solid mass near x=0. This is right. pictures
The example is written rather simply with Mathematica, a
language that supports an Interval data type (it took several
revisions of Mathematica to get it almost right.
Now it rounds up and down so
that 1/Interval[3.0,3.0] is
Interval[0.3...32, 0.33...34].
Can you guess its answer for Sin[Interval[1.0d60,1.0d60]]?.)