Search

13 Nov 2018: Supposethat there is a constant K such that ‖F(t,x) F(t,y)‖ K‖xy‖ for all t [0, 1] and all x,y B. ... Letx1,x2 Rn. By the Picard–Lindelöf theorem, we know that there is (0, 1] and differentiable functionsf1,f2 : [0,] B such that dfjdt = F(t,fj

20 Oct 2014: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)?

23 Oct 2017: b) If (fn) does not converge uniformly, show that there is a convergent sequence (xm) x [0, 1] such that (fn(xn)) does not converge to f(x). ... a) The space Cb(R) of bounded continuous functions on R.(b) The space C0(R) of continuous functions f : R R

12 Nov 2019: Show that φis isotopic to the identity. 8. Let α,β be a pair of simple closed curves on a surface S, such thati(α,β) = 1. ... Let α,β be astandard pair of simple closed curves on S such that i(α,β) = 1, andlet γ be the boundary curve.

21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

21 May 2005: 1. 2 ANALYSIS II EXAMPLES 1. (ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a,b] such that fn(xn) does ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

17 May 2019: Show. there exist disjoint open sets U1 and U2 such that Ci Ui.8. ... continuous path. f : [0, 1] Xconnecting a and b such that f is a homeomorphism onto its image.

23 Oct 2013: 11 fn 6 0. Is it possible to find such a sequence with |fn(x)| 1 for all n and for all x? ... On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6

2 Nov 2015: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f

1 Nov 2021: Fix such an n and prove the theorem as follows. (a) Show directly that a nowhere-zero vector field on Sn induces a homotopy from the identityto the antipodal map. ... b) Use de Rham cohomology to prove that no such homotopy exists.