8 - Sequences of Functions Flashcards
Pointwise and uniform convergence. Cauchy sequence of functions. Continuity of the limit function. Differentiability of real and complex functions.
What is pointwise convergence?
Pointwise convergence is one of the various senses in which a sequence of functions can converge to a particular function. It is weaker than uniform convergence, to which it is often compared.
We say the sequence of functions f_n converges pointwise to f on D if:
f(x) = lim(n->∞) f_n (x)
for all x ϵ D. We write f_n -> f pointwise.
Does pointwise convergence preserve continuity?
The limit function f in pointwise convergence is not continuous even though all functions f_n are continuous. Hence pointwise convergence does not preserve continuity.
What is uniform convergence?
A sequence of functions f_n converges uniformly to a limiting function f on a set D if, given any arbitrarily small positive number ε a number n0 can be found such that each of the functions f_n0, f_(n0+1), f_(n0+2), … differ from f by no more than ε at every point x in D.
We say f_n -> f uniformly on D if for every ε > 0 there exists n0 ϵ {N} such that
|f_n (x) - f(x)| < ε
for all n > n0 and all x ϵ D. We also sometimes say f_n (x) -> f(x) uniformly with respect to x ϵ D.
What is the comparison between Unifrom and Pointwise convergence?
Uniform convergence always implies pointwise convergence but not the other way around. The difference between uniform and pointwise convergence, for every ε > 0, the same n0 can be chosen for all x ϵ D such that
|f_n (x) - f(x)| < ε
is true. Graphically, this means that graph of f_n is within distance ε from the graph of f for all n > n0.
Is it possible to get uniform convergence from a pointwise convergent function?
Yes, if a sequence of functions does converge pointwise but not uniformly we can often restrict the domain such that we have uniform convergence on a smaller domain.
What is the Weierstrass M-Test?
The Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies to series whose terms are bounded functions with real or complex values and is analogous to the comparison test for determining the convergence of a series of real or complex numbers.
Suppose that (fn) is a sequence of real- or complex-valued functions defined on a set A and that there is a sequence of non-negative numbers (Mn) satisfying the conditions:
- |f_n ( x )| ≤ M_n for all n ≥ 1 and all x ∈ A
- ∞∑(n=1) M_n converges.
Then the series
∞∑(n=1) f_n (x)
converges absolutely and uniformly on A.
Relationship between continuity and uniform convergence
It can be used to show that a sequence of functions does not converge uniformly. If all functions f_n are continuous, f_n -> f pointwise, but f is not continuous, then f_n does not converge uniformly.
What is local uniform convergence?
We say that f_n -> f locally uniformly on D if for every x ϵ D there exists r > 0 such that f_n -> f uniformly on B(x,r) ∩ D
Does uniform convergence preserve continuity?
Yes, uniform convergence preserves continuity, and it also allows for interchangement of limit and integration
What is a sequence of functions?
A sequence that is made up of functions.
When is a power series continuous?
Any power series is continuous inside its disk of convergence.
Can integration and limit of a sequence of functions swap places if there is uniform convergence?
Yes
