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.
Page 5 of 5