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 e > 0 there exists a number N such that for all x in S, for all n > N , | fn (x) -f(x) | < e.

 

#4.Let for x Î 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 haveand £ ____.( ____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 Î 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 onR.

We must show: (the negation of the definition)

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

| fn (x) - f(x) | __(<,>,£, ³)e.

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

Then we have | fn (x) - f(x) | =______________________________________ (<,>,£, ³)1 = e.

 

(NOTE: In the _____________________substitute forfn (x) and f(x) and simplify, applying x = en.)

 

#6.Let for x Î [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.Letfor x Î [-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 Î.

 

 

 

 

 

 

#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 sequenceof 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.

 

(explanation not required)

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