Question

2. Let f:R + R and g: R + R be functions both continuous at a point ceR. (a) Using the e-8 definition of continuity, prove that the function f g defined by (f.g)(x) = f(x) g(x) is continuous at c. (b) Using the characterization of continuity by sequences and related theorems, prove that the function fºg defined by (f.g)(x) = f(x) · g(x) is continuous at c. (Hint for (a): try to use the same trick we used to prove that the product of two convergent sequences converges to the product of the limits)

Answer 1

I have done it for you in detail. Kindly go through.

gi RR be continuous functions 2. Let f: RR and point ceR. al a) Let E > be is c, Since g givers Since f is continuous at a, 75, so such that for ach and lx-cko,, If(x)-f(c) | < € Similarly, continuous at 3 8230 such that for xER and lx-clada, 1960)-g() < E. Now, Ifog ca) – fog cy) - I frav.ge) - fres.grey - fro.gra) – far.ge) + f().g6e) - frogas s fron] Igea) – gca) + 1gce)] 1 fran-fcos) Now for S= min {S, de o and holds. and I flee) = 1 f(x) = f(e) + fcco | < (fen) + 1f6)-fco If66) + E } for lx-cl<s. No if kell with lx-cl <S, Ifgew) - figcol = |frol1964) -gce)l+Igccsllfox) -960) <(lfacol +€) € + Igo) E - Clf(c)] + 1ģcc) +€) $

is com get the final of such {xn} converging to co This is enough to prove I f.ge)- figre) | <ke for some positive k. Thue here k- If(e)l + 1g6e)+€ >0. give fig continuous at x=c. Note that adjust & to need stef less than & instead of kɛ. But no unwanted cakulations. at a b) A function f:RR is continuous iff for every sequence to f(c). the {fland} converges Sequence the characterization of continuity by sequences. Suppose filRoR and g: R+R be continuous atc. To prove fig : RR is continuous Let {n} be a sequence converging a are continuons at {fons -rfic) and {gc«n)} — gce). ů, {fcam). gaxon]} fcc). bomb gcc). since ana → Canbo) ab. ie, {f.g (2n)} f.g (c). so by characterization of continuity applied to fig. get fig is continuous at c. at c. converging to c. Then since f and C, we