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3110 Let T be the ring of continuous functions from R to R and let f,g be given by f(x) = ˆ 0 if x ≤ 2 x−2 if 2 < x, g(x) = ˆ 2−x if x ≤ 2 0 if 2 < x Show that f,g ∈ T and that fg = 0 T, and therefore that T is not an integral domain Well, f and g are certainly continuous on the intervals (−∞,2) and (2,∞) since they are. ° g_gkgag fÂ/²&g>ßfÃh h hlhxh fþ /²&gg"0bg fÃgog5gqg=g" °g g f¹ e© ü>úgkgcgvg" e©/²&g µfþ>õ>üg0gwgzggg"gdgwg4gdg f÷0bg. G suc h that b ðx Þcðx Þb ðf ðx ÞÞ 1 ¼ c 0ðx Þ holds fo r all x A X Let c X !.

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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 x. G is called a G valued coc ycle G valued coc ycles c and c 0 are cohomo logous if there is a continu ous map b X !. Definiti on 21 Let ðX ;f Þ be a Canto r mini mal system and G be a nite gro up A conti nuous map c X !.

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