200
III - Convergence: Continuous variables
left and right do not always exist beyond the case of monotone or continuous
functions.
Here we must invoke the ghost of G. H. Hardy again: the symbols c + 0
and c - 0 do not denote the number c; if such were the case, one could
indeed deduce corollaries as to the mental equilibrium of mathematicians.
They have no meaning in themselves, and their sole legitimacy is to figure
in the context where they have just appeared. Some people have recently
invented the notation f (c+) and f (c- ), mainly so as not to "traumatise" the
little dears who, it appears, might believe that if one writes c+O rather than c,
this must be because in Transcendent Mathematics, as one sometimes called
it in the romantic age, c+O is not always equal to c. But in the presence ofthe
notation f(c+), these same will ask: "c plus what?"; one will have to reply:
"c plus nothing", whence the following question: "then why this + sign?",
and the dialogue will end in general hilarity. The only advantage that one
can acknowledge in the notation f(c+) is that it can be typed more quickly
than f(c + 0). Mathematical notations are what they are: purely writing
conventions 3 .
Let us add that both these notations have the drawback of irresistibly
suggesting that the function f is defined at the point c. This hypothesis is
not at all necessary as we have seen above, and as we shall see in the following
chapter when we define real powers. In this last case, one assumes that the
expression aX = f(x) is defined for a real> 0 and x E Q - it is an increasing
function of x if a > 1, decreasing if a < 1 - and seeks to define aX for x E JR
so that the new function, defined on JR, will again be monotone, in the same
sense as its restriction to Q. The method consists of observing that, for x ¢ Q,
the required number aX must lie between f(x - 0) and f(x + 0), and then
to prove that these two limit values are equal, to determine aX without any
ambiguity.
4 - The intermediate value theorem
The axiom of the existence of least upper bounds leads to a very simple
characterisation of intervals which we shall use immediately:
Theorem 5. For a set I C JR to be an interval it is necessary and sufficient
that
{(x E I) & (y E I) & (x < z < yn ===} z E I.
The condition is clearly necessary. To show that it is sufficient, consider
the greatest lower bound and least upper bound, possibly infinite, of I. For
any number z satisfying inf(I) < z < sup(I), with strict inequalities, there
exist, by definition, elements x and y of I such that x < z and z < y. Then
3 In this circle of ideas, we remark that some authors write x -+ c_ for what we
write as x --+ C, x < c. One might also write x --+ c - o.
III - Convergence: Continuous variables
left and right do not always exist beyond the case of monotone or continuous
functions.
Here we must invoke the ghost of G. H. Hardy again: the symbols c + 0
and c - 0 do not denote the number c; if such were the case, one could
indeed deduce corollaries as to the mental equilibrium of mathematicians.
They have no meaning in themselves, and their sole legitimacy is to figure
in the context where they have just appeared. Some people have recently
invented the notation f (c+) and f (c- ), mainly so as not to "traumatise" the
little dears who, it appears, might believe that if one writes c+O rather than c,
this must be because in Transcendent Mathematics, as one sometimes called
it in the romantic age, c+O is not always equal to c. But in the presence ofthe
notation f(c+), these same will ask: "c plus what?"; one will have to reply:
"c plus nothing", whence the following question: "then why this + sign?",
and the dialogue will end in general hilarity. The only advantage that one
can acknowledge in the notation f(c+) is that it can be typed more quickly
than f(c + 0). Mathematical notations are what they are: purely writing
conventions 3 .
Let us add that both these notations have the drawback of irresistibly
suggesting that the function f is defined at the point c. This hypothesis is
not at all necessary as we have seen above, and as we shall see in the following
chapter when we define real powers. In this last case, one assumes that the
expression aX = f(x) is defined for a real> 0 and x E Q - it is an increasing
function of x if a > 1, decreasing if a < 1 - and seeks to define aX for x E JR
so that the new function, defined on JR, will again be monotone, in the same
sense as its restriction to Q. The method consists of observing that, for x ¢ Q,
the required number aX must lie between f(x - 0) and f(x + 0), and then
to prove that these two limit values are equal, to determine aX without any
ambiguity.
4 - The intermediate value theorem
The axiom of the existence of least upper bounds leads to a very simple
characterisation of intervals which we shall use immediately:
Theorem 5. For a set I C JR to be an interval it is necessary and sufficient
that
{(x E I) & (y E I) & (x < z < yn ===} z E I.
The condition is clearly necessary. To show that it is sufficient, consider
the greatest lower bound and least upper bound, possibly infinite, of I. For
any number z satisfying inf(I) < z < sup(I), with strict inequalities, there
exist, by definition, elements x and y of I such that x < z and z < y. Then
3 In this circle of ideas, we remark that some authors write x -+ c_ for what we
write as x --+ C, x < c. One might also write x --+ c - o.