Already the simplest of examples make it clear, that it is
important to study real (valued) functions f which are not
defined for every real argument. Thus it is appropriate to
associate to every real function f its domain D(f). In the
following, let x range over all real numbers. As f may not be
defined for x , it must be made explicit that x is in D(f) when
statements are considered which involve values fx .
If D(f) is an open interval, then continuity of f at a point a of
D(f) is characterized by the condition
(1) for every e>0 there is a d>0 such that for every x :
if |x - a| < d then x in D(f) and |fx - fa| < e .
This characterization remains in effect if, instead of being an
interval, D(f) is at least open, i.e. if
(o) for every x in D(f) there is an open intervall containing x
and contained in D(f) .
Conversely, if f is such that (1) holds for every a in D(f) then
D(f) is open.
If D(f) is arbitrary, then continuity of f at a is characterized by
(2) for every e>0 there is a d>0 such that for every x :
if |x - a| < d and x in D(f) then |fx - fa| < e ,
and this is equivalent to
for every e>0 there is a d>0 such that for every x in D(f):
if |x - a| < d then |fx - fa| < e .
The condition (2) - in a slightly different wording - occurs, for
instance, as definition of continuity in E.W.Hobson's "The theory
of functions of a real variable", vol.I , Cambridge UP 1907/1921/1927
(in the third edition, reprinted by Dover, on p.281).
Authors of books written not for students of mathematics, but for
future engineers, biologists etc., often believe it to be didactically
useful not to confront their readers with the 'more difficult'
notion of real functions with arbitrary domains, and so they
define continuity only for functions for which D(f) is an open
interval, or is at least an open set; hence they only will use
(1) and not the 'more difficult' condition (2). Hardy, it appears,
proceeded in this manner in his "A Course of Pure Mathematics",
e.g. in the 1952 edition on p.186 . But also authors writing for
students of mathematics have introduced continuity in two steps,
for instance M.H.Protter and C.B.Morrey in "A First Course in
Real Analysis", Springer 1977 . On p.31 they first define f to be
continuous at a if it satisfies both (o) and (1), and on p.44
they extend their definition, calling f continuous if it
satisfies (2). A few lines later, they observe that a function is
44-continuous at every isolated point of D(f) [while at such a
point it will not be 31-continuous since it violates (o), hence
also the conjunction "(o) and (1)"].
(II)
The conditions "(o) and (1)" or "(2)" single out, within the
class of all real functions, subclasses of continuous functions;
this fits the scholastic rule "definitio fit par genus proximum
[here the class of real functions] et differentiam specificam".
In the case of the continuous functions, however, most didactically
inspired authors, Hardy among them, do not want to use a differentia
specifica expressed as a conjunction; they prefer to use instead
a narrower genus proximum, namely the class of real functions
satisfying (o), and then they define their continuous functions
by singling them out through the specific difference (1). Of
course, both of the approaches determine as their result the same
class of 31-continuous or 186-continuous functions, but in the
second approach the condition (o) appears as a delimiting, overall
hypothesis, under which the selection according to (1) takes place.
In everyday's language, we conceive a hierarchy of genera, beginning
with the all-compassing "everything" and going down through, say,
"mathematical objects" and "real functions", to the two genera of
"[44-]continuous functions" and "real functions satisfying (o)"
which have as their intersection the 31-continuous functions.
Clearly, if A, B, C are genera and if
B is defined in A by a property p(j)
C is defined in B by a property q(j)
then C may also be seen as defined in A by a property r(j),
namely the conjunction "p(j) and q(j)". In particular, any genus
C may be seen as defined in the all-compassing "everything" by a
sufficiently long conjunction s(j), formed from the defining
properties of the intermediary genera.
(III)
The point of view about the genus proximum matters when it comes
to formulate negations. When stating that some g has the property
"not-p(j)" for a certain property p(j), this always includes the
(at least implicit) reference to the genus proximum A from which
g comes and within which p(j) is conceived to select. Thus in the
above A-B-C-situation, the alternative
Q any b in B either has the property q(j) or the property
not-q(j)
refers to the definition of C in B , while the alternative
P any a in A either has the property r(j) or the property
not-r(j)
refers to the definition of C in A with r(j) as "p(j) and q(j)",
and the alternative
S any a either has the property s(j) or the property not-s(j)
refers to the definition of C in the genus "everything" by a
suitable property s(j).
Consider now the particular case that
A is the class of all real functions,
B is the class of functions in A satisfying the property p(f)
expressed in (o), and
C is the class of functions in B satisfying the property
q(f) expressed in (1) ,
i.e. the class of 186-continuous functions. Let f be the function
which has as D(f) the set of rational numbers and which is
constant, having the same value for all of its arguments. Then f
is 44-continuous at each of its arguments, and for no x in D(f)
the condition (o) is satisfied. If with Protter and Morrey we
consider C as defined within A, then by alternative P the
function f is not-31-continuous. If, however, with Hardy and
other authors, we consider C as defined within B, then the
alternative Q does not apply as f is not in B, and so f is
neither 186-continuous nor not-186-continuous.
Thus there remains the oddity in Hardy's book that during his
discussion he silently changes his point of view. In defining his
186-continuous functions he uses B as his frame of reference,
whereas in the exercise 2 he uses A when he writes that f is not
31-continuous. Unfortunately, Professor Hardy cannot be reached
for the time being, but I am certain that, should we once ask him
about his apparent inconsistence, he would take down his glasses,
smile at us impishly, and then reply "but didn't you learn to
argue carefully from my formulations ? ". Which we could not deny.
W.F.