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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 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 Ã Let f (x) = 0.
Complete the following discussion and proof that (fn) converges uniformly to f on .
Suppose e is any positive real number.
We want to find N such that for all x Ã and n > N, we have |fn(x) -f (x)| =< e
Note that since x Â³ we haveand Â£ ____.( ____are numbers in simplest form)
______for all x Ã
(___ is an expression involving an appropriate constant and the variable n only, no x)
So, we want______< e, which implies that n > _____.
Let > 0. Choose N = _____. For all x Ã , and n > N, we have
______ <_______= e, as desired.
(______ should be the expression involvinge, before being simplified to get exactly e.)
#5.Let for x Ã 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 inRand _____ (all/some) n> N ,
| fn (x) - f(x) | __(<,>,Â£, Â³)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) | =______________________________________ (<,>,Â£, Â³)1 = e.
(NOTE: In the _____________________substitute forfn (x) and f(x) and simplify, applying x = en.)
#6.Let for x Ã [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 Ã [-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 Ã.
#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)