Math HW help with functions

Sequences of Functions

Def. Let (fn) be a sequence of functions defined on a subset S of R. Then (fn) converges uniformly on S to a function f defined on S if

For each  > 0 there exists a number N such that for all x in S, for all n > N , | fn (x)  f(x) | < .

#4. Let for x  Let f (x) = 0.

Complete the following discussion and proof that (fn) converges uniformly to f on .

Discussion:

Suppose e is any positive real number.

We want to find N such that for all x Î and n > N, we have |fn(x) - f (x)| = < e

Note that since x ³ we have and £ ____. ( ____ are numbers in simplest form)

______ for all x Î

(___ is an expression involving an appropriate constant and the variable n only, no x)

So, we want ______ < e, which implies that n > _____.

Proof:

Let > 0. Choose N = _____. For all x Î , and n > N, we have

______ < _______ = e, as desired.

(______ should be the expression involving e, before being simplified to get exactly e.)

#5. Let for x  R. Let f(x) = 3x2.

Clearly, (fn) converges pointwise to f. But does it converge uniformly to f?

Fill in the blanks to carefully show that (fn) does not converge uniformly to f on R.

We must show: (the negation of the definition)

For _____ (all/some) e > 0, for _____ (all/some) N, for _____ (all/some) x in R and _____ (all/some) n > N ,

| fn (x)  f(x) | __ (<,>,, ) e.

Let = 1. Given any N , let n be a positive integer greater than N, and set x = en.

Then we have | fn (x)  f(x) | = ____________________________________ __ (<,>,, ) 1 = e.

(NOTE: In the _____________________ substitute for fn (x) and f(x) and simplify, applying x = en.)

#6. Let for x  [0, 1].

#6(a) State f (x) = lim fn(x).

#6 (b) Determine whether (fn) converges uniformly to f on [0, 1]. Carefully justify your answer, either showing that the definition (shown on the previous page) holds, or proving that it cannot hold.

#7. Let for x  [0.8, 1].

#7 (a) State a formula for f (x) = lim fn(x). (no explanation required)

#7 (b) (fn) does not converge uniformly to f on [0.8, 1]. How can you deduce this fact without working hard? Please explain. HINT: Apply an appropriate theorem and an observation about the function f(x) you found in part (a).

Series of Functions (#8, 12 pts)

#8. Determine whether or not the given series of functions converges uniformly on the indicated set. Justify your answers with work/explanations . (HINT: Consider the Weierstrass M-Test and find an appropriate sequence Mn )

#8 (a) for x in R.

#8(b) for x .

#9. Show that it is possible to have a sequence of functions that are discontinuous at every point yet converge uniformly to a function that is continuous everywhere.

That is, state an example of a sequence of functions (fn) and a function f satisfying all of the following:

• Each fn is discontinuous at every real number.

• (fn) converges uniformly to f .

• f is continuous at every real number.

Let g: [0,1] -> IR g(x) =1 if x<Q, =0 else

[g is called Dirichletian Function]

Set, fn(x) =g(x)/n for all x [0,1]

Since g is discontinuous at every real number in [0,1], fn’s are discontinuous everywhere.

Let f(x) = 0 for every x G [0,1]

Than, {fn} converge uniformly to f and f is continuous everwhere.