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2 (a) Define uniform continuity on R for a function f R → R (b) Suppose that f,g R → R are uniformly continuous on R (i) Prove that f g is uniformly continuous on R (ii) Give an example to show that fg need not be uniformly continuous on R Solution • (a) A function f R → R is uniformly continuous if for every ϵ > 0 there exists δ > 0 such that f(x)−f(y) < ϵ for all xЃx X g p g i Y _ ސ쌧 c s R P Q U TEL FAXN r S A F m lj P B x X g P A ̂v d a T C g ʌ n s ˉz @ r101




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X(g)Y(g)↽−−⇀Z(g)𝐾p=100 at 300 KX(g)Y(g)↽−−⇀Z(g)Kp=100 at 300 K In which direction will the net reaction proceed for the initial conditions X=Y=Z=10 M?X=Y=Z=10 M?♫ ROMIJUVN x GPR Beat Karma (Monsterwolf Free Release) Monsterwolf Release ♫ Trap Music / Mix Download Mp3 https//fanlinkto/romijuvnxgprbeatX,g 1h2G x,g 1hx= x,hx= gx In fact, this argument read backwards, shows also that the map is onetoone The map is clearly onto Finally, h(gG x) = (hg)G x7!(hg)x= h(gx) so the mapping is consistent with the two actions Corollary If Gis a nite group acting on a set X, then every orbit is a nite set and its cardinality divides the order




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(a) f is onetoone iff ∀x,y ∈ A, if f(x) = f(y) then x = y (b) f is onto B iff ∀w ∈ B, ∃x ∈ A such that f(x) = w (c) f is not onetoone iff ∃x,y ∈ A such that f(x) = f(y) but x 6= y (d) f is not onto B iff ∃w ∈ B such that ∀x ∈ A, f(x) 6= w 2 For each of the following, give an example of sets A, B




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