
Draw the graph of y = sin (1/x)
for x < 1.
Color blue the regions inside the graph
where y is positive, and
red those where it is negative.
The basic way to draw a graph of a function is by
drawing it as a sequence of small
lines between (x, f(x)) and (x+dx, f(x+dx)).
There are several things that make this graph not quite ordinary.
The function 1/x is singular at 0, so you cannot just graph
an entire interval
through 0, and must plot left and right separately.
But then the graph oscillates at a higher and higher frequency
as it gets close to x=0,
with period decreasing below any fixed size. This requires that
the interval dx decrease as the graph gets closer to 0.
How should it be chosen? Roughly speaking, the number of
lines in each loop of the graph should remain constant,
say N = 100. The graph crosses
the xaxis at the points 1/ π n, so one loop
lasts between 1/ π n and 1/ π (n+1).
What the program belwo does is plot each loop separately.
Another possible
solution would be to set dx = x^{2}/N, since
1/ π n  1/ π (n+1) = 1/ π n(n+1) which is roughly proportional
to x^{2} if x = 1/ π n. (This is not what I thought
originally.) Doing this makes it easy to color alternate loops
in different colors.
