Example 6.2.1: For each n, define fₙ : [0,1] → ℝ by
fₙ(x) = 1 - n x when x < 1 / n and fₙ(x) = 0 when x ≥ 1 / n.
Each fₙ is continuous, the pointwise limit is the function that is 1
at 0 and 0 for x > 0, and this limit is not continuous at 0.
Instances For
The pointwise limit function in Example 6.2.1, equal to 1 at 0 and
0 elsewhere.
Instances For
The sequence example6_2_1_fn n consists of continuous functions.
For any x > 0, the sequence example6_2_1_fn n x converges to 0.
At x = 0, the sequence example6_2_1_fn n 0 converges to 1.
The pointwise limit of example6_2_1_fn on [0, 1] is the function that
is 1 at 0 and 0 elsewhere, and this limit function is not continuous at
0.
Theorem 6.2.2: If fₙ : S → ℝ is a sequence of continuous functions that
converges uniformly to f : S → ℝ, then f is continuous.
Example 6.2.3: The sequence of piecewise linear functions
fₙ : [0,1] → ℝ given by fₙ(0) = 0, fₙ(x) = (n + 1) - (n + 1)² x
for 0 < x < 1 / (n + 1), and fₙ(x) = 0 for x ≥ 1 / (n + 1) is
Riemann integrable with integral 1 / 2. The pointwise limit on [0,1]
is the zero function, so lim_{n → ∞} ∫₀¹ fₙ = 1/2 while
∫₀¹ (lim_{n → ∞} fₙ) = 0.
Equations
Instances For
Each example6_2_3_fn n has integral 1/2 on the interval (0, 1].
For any x ∈ [0,1], the sequence example6_2_3_fn n x converges to 0.
The integrals of example6_2_3_fn n converge to 1/2, while the
integral of the pointwise limit is 0, showing that pointwise convergence
alone does not allow interchanging limit and integral.
Theorem 6.2.4: If fₙ : [a, b] → ℝ is a sequence of Riemann integrable
functions converging uniformly to f : [a, b] → ℝ, then f is Riemann
integrable and the integral of the limit equals the limit of the integrals.
Example 6.2.5: For fₙ(x) = ((n x) + sin (n x²)) / n on [0,1], the
functions converge uniformly to x, so the integrals converge to the
integral of the limit. Therefore
lim_{n → ∞} ∫₀¹ ((n x + sin (n x²)) / n) = 1 / 2.
Example 6.2.6: The functions fₙ : [0, 1] → ℝ are 1 at rationals
whose reduced denominator is at most n and 0 otherwise. Each fₙ is
Riemann integrable with integral 0, the sequence converges pointwise to the
Dirichlet function that is 1 on ℚ and 0 on irrationals, and this limit
is not Riemann integrable.
Instances For
Each function in the sequence from Example 6.2.6 is Riemann integrable on
[0, 1] with integral 0.
The sequence example6_2_6_fn n converges pointwise to the Dirichlet
function on [0, 1].
The Dirichlet function on [0, 1] is not Riemann integrable.
Example 6.2.7: The functions fₙ : [0,1] → ℝ are given by
fₙ(x) = 0 for x < 1 / (n + 1) and fₙ(x) = 1 / x otherwise.
Each fₙ is bounded on [0, 1] by n + 1, but the pointwise limit
f(x) = 0 for x = 0 and f(x) = 1 / x otherwise is unbounded.
Instances For
The sequence in Example 6.2.7 converges pointwise on [0, 1] to the
function that is 0 at 0 and 1 / x elsewhere.
The pointwise limit in Example 6.2.7 is unbounded on [0, 1].
Example 6.2.8: The functions fₙ(x) = sin ((n + 1) x) / (n + 1) converge
uniformly to 0, whose derivative is 0. The derivatives
fₙ'(x) = cos ((n + 1) x) do not converge pointwise: at π they oscillate
between 1 and -1, and at 0 they are constantly 1, not approaching 0.
Instances For
The sequence from Example 6.2.8 converges uniformly to the zero function.
At π, the derivatives in Example 6.2.8 oscillate and do not converge.
At 0, the derivatives in Example 6.2.8 are constantly 1, so the sequence
does not approach 0.
Example 6.2.9: Let fₙ(x) = 1 / (1 + n x²). For x ≠ 0, fₙ(x) converges
to 0, while fₙ(0) converges to 1, so the pointwise limit is not
continuous at 0. The derivatives are fₙ'(x) = - 2 n x / (1 + n x²)², which
converge pointwise to 0 but not uniformly on any interval containing 0.
The limit function fails to be differentiable at 0.
Instances For
Pointwise limit function in Example 6.2.9, equal to 1 at 0 and 0
elsewhere.
Instances For
For x ≠ 0, example6_2_9_fn n x converges to 0 as n → ∞.
At x = 0, the sequence example6_2_9_fn n 0 converges to 1.
The pointwise limit of Example 6.2.9 is not continuous at 0.
For every x, the derivatives in Example 6.2.9 converge pointwise to 0.
The derivatives from Example 6.2.9 do not converge uniformly on any
interval containing 0.
The pointwise limit in Example 6.2.9 is not differentiable at 0.
Theorem 6.2.10: Let I be a bounded interval and let fₙ : I → ℝ be
continuously differentiable functions. If the derivatives fₙ' converge
uniformly on I to g and the sequence of values fₙ(c) converges for some
c ∈ I, then fₙ converges uniformly on I to a continuously differentiable
function f with derivative f' = g.
Proposition 6.2.11: A power series ∑ cₙ (x - a)^n with radius of
convergence ρ > 0 converges uniformly on [a - r, a + r] for every
0 < r < ρ. As a consequence, the sum defines a continuous function on the
open interval where the series converges (or on all of ℝ if ρ = ∞).
Corollary 6.2.12: For a convergent power series
∑_{n=0}^∞ cₙ (x - a)^n with radius of convergence ρ > 0, if I is the
interval (a - ρ, a + ρ) (or ℝ when ρ = ∞) and f denotes the limit,
then for every x ∈ I one has
∫ₐˣ f = ∑_{n=1}^∞ (c_{n-1} / n) (x - a)^n, and the radius of convergence of
the latter series is at least ρ.
Corollary 6.2.13: For a convergent power series
∑_{n=0}^∞ cₙ (x - a)^n with radius of convergence ρ > 0, let
I = (a - ρ, a + ρ). If f is its sum on I, then f is differentiable on
I and f' (x) = ∑_{n=0}^∞ (n + 1) c_{n+1} (x - a)^n, whose radius of
convergence is also ρ.
Example 6.2.14: The power series ∑ x^n / n! has infinite radius of
convergence, so it defines a function f(x) = ∑ x^n / n! on all real
numbers. It satisfies f(0) = 1, differentiates term by term to give
f' = f, and coincides with the usual exponential function.