Graphics Reference
In-Depth Information
3
Continuity: when using a pencil to make a 'continuous' line the
Intermediate Value Theorem appears to be unfalsifiable. This is
closely related to the denseness
—
completeness issue in 2.
4
What a continuous function may belikeis not obvious. A function
may be constructed, somewhat unexpectedly, from the segment
[0, 1] onto theunit square
(
x
,
y
)
0
x
1, 0
y
1
constructed by mapping the number 0.
a
a
a
a
a
a
...tothe
point (0.
a
...).
This function is well defined provided terminating decimals in
[0, 1] are always represented by recurring 9s. The function is
continuous and one-to-one except where one of the image
coordinates terminates.
a
a
. . ., 0.
a
a
a
5
The connection between continuity and differentiability seems
straightforward enough (with non-differentiability at the occasional
sharp point on the curve) until one considers a function which is
everywhere continuous and nowhere differentiable, such as David
Tall's
Blancmange Function
in qn 12.46.
The examples 2
—
5 above all show the inadequacy of considering a mark
made by a pencil, without lifting it from the paper, as the illustrative
model of the graph of a continuous function. It is tantalising to contrast
these illustrations of suspect visualisation with an example where the
intuitive, pencil and paper, point of view is sound enough, as at the
beginning of chapter 7. Try finding the kind of set
A
which can bethe
rangeof a continuous function R
A
R. Here even quite rough work
with pencil and paper leads to a precise and accurate formulation.
I would certainly concur with the judgement of J. E. Littlewood
who wrotein 1953 (
Bolloba´ s
, 1986, p. 54):
My pupils
will
not use pictures, even unoMcially and when
there is no question of expense. This practice is increasing.
I have lately discovered that it has existed for 30 years or
more, and also why. A heavy warning used to be given
that pictures are not rigorous; this has never had its bluff
called and has permanently frightened its victims into
playing for safety. Some pictures, of course, are not
rigorous, but I should say most are(and I usethem
wherever possible myself). An obviously legitimate case is
to usea graph to definean awkward function (
e
.
g
.
behaving differently in successive stretches).
