Question

2. Let DCR, and suppose the functions f:D + R and g: D R are continuous at to ED. Use the - definition of continuity to prove that f+g and fg are both continuous at ro e D: that is, prove that for every e > 0 there exists 8 >0 such that (+9)(2)-(+9)(20) <and (9)(x) - (49)(20) << whenever re D and r-rol < 8. (Hint. Use inequalities similar to those we used to prove the cor- responding results for sums and products of convergent sequences. In the case of fg, also use #1 above.)

Answer 1

(3 solution : I given that and 9: D-R are fib-7 continuous - at HOED then for every E, SO 7830, sit If(x-flood l <f, when even lx-xole8,- and for every foo & 20 sit 19 (4) – g(he) <Ę when even fat-Xole & - © Partil . To continuous at do ED.. prove ft. is Let 8 = min {8, E) and for Exo, Site, E = E/

1 4 e becomes sfees - f(d) <wheneven (7-1466) and 19(x) - g(x) <%. Wheneven (x-odoje Now, if(g) (x) = f(x) +90x] [:$¢g areccntioung & HED :14( 9) (4) (Pty) = pcu )+ 902) - Plan) g(xo) 3. fl cot) - ACxe))+964) -900 <ifax) - ffxco) 1+1 908) - geed can e Ele + fle Pusong Catollalk 1bd] => \f + g) 0 - (f+glxo) <€ when (x-xa) 26 . y fty is continuous at doen . Hence Proved.

Part 20 fg is continuous at XoED Hence Afg) (x) = f(x) g(x) KYED (1. fg a reconi uong => 168g)(x) - 89) (k)- ford g(x) – Flxo) 910 -> fcd) g(x) = g(u flog]+ g(x)=f(dal-ffXo]9CXO) -> 90 f(x) = f(x) + f ) (9685-9 (0)] / $(a)) ifCN) - P(x2)+ 1468) | 1961-9(%)).c from ③ we have Ifa) - f(t) <E, wheneven (2-xo] <6, choose E, = for even txo facit go) => Ifa) - f(x) < = when even .. 2114968) (x-xod eo -☺

ther frow ③ we have Igx -g(20) <tz whenever Ix-xo) <62 Choose G = G 21.6CX01) => (918 - g(xol at wwern (x-to-&-6 21ffro Also 3 ) - 94 - (+) + xa) < 340 - 34 / 1369 FE2+19(801) choose fq=1 wheneren ] 6₂ >0 sit (a=Xolca 1961)) 1+ 19 () / ichen even (2-4) (83–@ . Let 8 = men ( 8 , &r, s}

Then (u) becomes If (9) (0) - POJ (do) < 1gW1f2 -f(x) + f(xo]| g(u)- g(x) / <li+19(80) J I f () - ffto) 14lf card (90-9 (80)/ vonig los pito (o) E lftal e 249460)) ankety Pesong 6) 401] - 1149)(4) –f(g) (Ho)/ when Ix-20/66 > fg is continuous at so E D Hence proved .. سے