588 
GEOMETRIC NOTIONS 
were i2-integrable. Then g 2 = 1 +f'(x) 2 is iibintegrable, and 
hence f'(x) 2 also. But the points of discontinuity of f' 2 in 51 do 
not form a null set. Hence/' 2 is not iü-integrable. 
On the other hand, Volterra’s curve is rectifiable by 569, 2, and 
528, l. 
572. Taking the definition of length given in 569, 1, we saw 
that the coordinates 
s = 0(0 > y = -0(0 
must have limited variation for the curve to be rectifiable. But we 
have had many examples of functions not having limited variation 
in an interval 5i. Thus the curve defined by 
y — x sin - , x 0 
x 
= 0 , x= 0 
(4 
does not have a length in Si = ( — 1, 1) ; while 
y = x 2 sin - , x =£ 0 
x (5 
= 0 , x = 0 
does. 
It certainly astonishes the naïve intuition to learn that the 
curve 4) has no length in any interval 8 about the origin how 
ever small, or if we like, that this length is infinite, however small 
8 is taken. For the same reason we see that 
No arc of Weierstrass' curve has a length (or its length is infinite') 
however near the end points are taken to each other, ivhen ab > 1. 
573. 1. 6° Property. Space-filling Curves. We wish now to 
exhibit a curve which passes through every point of a square, i.e. 
which completely fills a square. Having seen how to define one 
such curve, it is easy to construct such curves in great variety, not 
only for the plane but for space. The first to show how this may 
be done was Peano in 1890. The curve we wish now to define is 
due to Hilbert. 
We start with a unit interval 51 = (0, 1) over which t ranges, 
and a unit square 53 over which the point x, y ranges. We define 
x=4>(fi) , y = yjr(t) (1