III - Convergence: Continuous variables
§1. The intermediate value theorem - §2. Uniform convergence -
§3.
Bolzano- Weierstrass
and
Cauchy's
criterion
§4. Differentiable functions - §5. Differentiable functions of several variables
§1. The intermediate value theorem
1 - Limit values of a function. Open and closed sets
The concept of a limit value of a function defined on a set X C elements are not necessarily integers has already been introduced in nO 4 of
Chap. II, but we have hardly used it up to now. In this chapter we shall
develop it much further.
(a) Limit as x ----+ +00. Here we suppose that X C IR is not bounded
above. We then say that f(x) tends to a limit u as x tends to +00 if, for all
r > 0,
(1.1)
d[u, f(x)] < r for all sufficiently large x E X,
i.e. for x > N where in general N depends on r. We use the notation
u = lim f(x)
x-+oo
xEX
omitting "x E X" when there is no likelihood of ambiguity.
For example, for any integer n > 1,
lim l/x n = 0
x-++oo
since l/x n < l/x for x > 1, whence 11/xnl < r for x > max(l, l/r).
(b) Limit as x ----+ -00. Almost identical definition. In decimal language:
for every n E N there exists apE N such that