- Suppose is a sequence of decreasing positive functions defined on [0 ,∞) which converges pointwise to 0 for every x ∈ [0,∞). Can it be concluded that this sequence converges uniformly to 0 on [0,∞)? Now replace [0,∞) with. What can be said in this case assuming pointwise convergence still holds?
- If andare sequences of functions defined on D which converge uniformly, show that if a,b are constants, then af
_{n}+bg_{n}also converges uniformly. If there exists a constant, M such that,< M for all n and for all x ∈ D, showconverges uniformly. Let f_{n}≡ 1∕x for x ∈and let g_{n}≡∕n. Showconverges uniformly onandconverges uniformly butfails to converge uniformly. - Show that if x > 0,∑
_{k=0}^{∞}converges uniformly on any interval of finite length. - Let x ≥ 0 and consider the sequence . Show this is an increasing sequence and is bounded above by ∑
_{k=0}^{∞}. - Show for every x,y real, ∑
_{k=0}^{∞}converges and equals - Consider the series ∑
_{n=0}^{∞}^{n}. Show this series converges uniformly on any interval of the form. - Formulate a theorem for a series of functions which will allow you to conclude the infinite series is uniformly continuous based on reasonable assumptions about the functions in the sum.
- Find an example of a sequence of continuous functions such that each function is
nonnegative and each function has a maximum value equal to 1 but the sequence of
functions converges to 0 pointwise on .
- Suppose is a sequence of real valued functions which converges uniformly to a continuous function f. Can it be concluded the functions f
_{n}are continuous? Explain. - Let hbe a bounded continuous function. Show the function f= ∑
_{n=1}^{∞}is continuous. - Let S be a any countable subset of ℝ. Show there exists a function f defined on ℝ which is discontinuous at every point of S but continuous everywhere else. Hint: This is real easy if you do the right thing. It involves Theorem 6.9.15 and the Weierstrass M test.
- By Theorem 6.10.3 there exists a sequence of polynomials converging uniformly to
f=on the interval. Show there exists a sequence of polynomials,converging uniformly to f onwhich has the additional property that for all n,p
_{n}= 0 . - If f is any continuous function defined on , show there exists a series of the form ∑
_{k=1}^{∞}p_{k}, where each p_{k}is a polynomial, which converges uniformly to f on. Hint: You should use the Weierstrass approximation theorem to obtain a sequence of polynomials. Then arrange it so the limit of this sequence is an infinite sum. - Sometimes a series may converge uniformly without the Weierstrass M test being
applicable. Show
converges uniformly on

but does not converge absolutely for any x ∈ ℝ. To do this, it might help to use the partial summation formula, 5.6. - Suppose you have a collection of functions S ⊆ Cwhich satisfy
where γ ≤ 1. Show there is a uniformly convergent subsequence of S which converges uniformly to some continuous function. The second condition on f is called a Holder condition and such functions are said to be Holder continuous. These functions are denoted as C

^{0,γ}and this little problem shows that the embedding of C^{0,γ}into Cis compact. - Suppose f ∈ Cand
for every n ≥ 0, such that n is an integer. Show that then f

= 0 for all x. Hint: Use Weierstrass approximation theorem.

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