198
III - Convergence: Continuous variables
by f on E; and this is so. If u is finite, then for any r > 0 there exists a
c' E E such that f(c') > u - r; we have c' < c whence, as in Chap. II,
u - r < f(x) ::; u for all x E X such that c' < x < c,
i.e. for all x E E sufficiently close to c. So clearly f tends to u. The case
where u = +00 is even more obvious: for any M E JR, there is a c' < c in E
such that f(c') > M, whence f(x) > M for c' < x < c. In conclusion:
Theorem 4. Let X be a subset of JR, let f be a real function defined and
increasing on X, and let c be an adherent point of the set E of x E X such
that x < c. Then f(x) tends to a limit when x E X tends to c remaining < c,
and
(3.1)
lim f(x) = sup(f(E)) ::; +00.
x--+c,x
but we shall use it to define real powers in the next chapter, in the case where
X = Q. The fact that the point c has been deleted from E is essential. If we
choose E to be the set of x E X such that x ::; c, then to say that f(x) tends
to a limit as x E E tends to c would mean that the restriction of f to the
set E was continuous at the point c as we saw at the start of nO lor, if one
prefers, that f(x) tends to f(c) as x tends to c remaining < c; one says then
that f is continuous on the left at the point c. But there is no reason for this
to happen: consider on X = [0,1] the function equal to 0 for x < 1/2 and to
1 for x 2:: 1/2 and take c = 1/2: when x tends to 1/2 remaining in [0,1/2[, it
tends to 01= f(c), but has no limit when x tends to 1/2 in the closed interval
[0,1/2]. Square waves are similar.
We have supposed, in the preceding theorem, that x tends to c remaining
< c; there is an analogous statement for the case where x remains > c,
replacing E by the set of x E X such that x > c (supposing c adherent to
this set, i.e. a limit of points of X all > c).
Irrespective of Theorem 4, it frequently happens that given a function,
monotone or not, defined on a set X C JR of which a and b are the lower and
upper bounds, one is interested in what happens on a neighbourhood of a
point c E la, b[, so that there are two cases to envisage if c is adherent to the
two subsets of X defined by x < c and x > c (the case of an interval or of
Q, for example). In the first case, the limit, if it exists, of f(x) when x tends
to c remaining strictly smaller than c is traditionally denoted by
(3.2')
f(c - 0) =
lim f(x),
x--+c,x
similarly by
III - Convergence: Continuous variables
by f on E; and this is so. If u is finite, then for any r > 0 there exists a
c' E E such that f(c') > u - r; we have c' < c whence, as in Chap. II,
u - r < f(x) ::; u for all x E X such that c' < x < c,
i.e. for all x E E sufficiently close to c. So clearly f tends to u. The case
where u = +00 is even more obvious: for any M E JR, there is a c' < c in E
such that f(c') > M, whence f(x) > M for c' < x < c. In conclusion:
Theorem 4. Let X be a subset of JR, let f be a real function defined and
increasing on X, and let c be an adherent point of the set E of x E X such
that x < c. Then f(x) tends to a limit when x E X tends to c remaining < c,
and
(3.1)
lim f(x) = sup(f(E)) ::; +00.
x--+c,x
but we shall use it to define real powers in the next chapter, in the case where
X = Q. The fact that the point c has been deleted from E is essential. If we
choose E to be the set of x E X such that x ::; c, then to say that f(x) tends
to a limit as x E E tends to c would mean that the restriction of f to the
set E was continuous at the point c as we saw at the start of nO lor, if one
prefers, that f(x) tends to f(c) as x tends to c remaining < c; one says then
that f is continuous on the left at the point c. But there is no reason for this
to happen: consider on X = [0,1] the function equal to 0 for x < 1/2 and to
1 for x 2:: 1/2 and take c = 1/2: when x tends to 1/2 remaining in [0,1/2[, it
tends to 01= f(c), but has no limit when x tends to 1/2 in the closed interval
[0,1/2]. Square waves are similar.
We have supposed, in the preceding theorem, that x tends to c remaining
< c; there is an analogous statement for the case where x remains > c,
replacing E by the set of x E X such that x > c (supposing c adherent to
this set, i.e. a limit of points of X all > c).
Irrespective of Theorem 4, it frequently happens that given a function,
monotone or not, defined on a set X C JR of which a and b are the lower and
upper bounds, one is interested in what happens on a neighbourhood of a
point c E la, b[, so that there are two cases to envisage if c is adherent to the
two subsets of X defined by x < c and x > c (the case of an interval or of
Q, for example). In the first case, the limit, if it exists, of f(x) when x tends
to c remaining strictly smaller than c is traditionally denoted by
(3.2')
f(c - 0) =
lim f(x),
x--+c,x
similarly by